Good question! I think to start off the discussion I'd say this falls into the "similar-but-different" category-- though both were kings who were executed by their people, the instances were quite differenct, as was the greater context that the event took place within!

I talked about this in relation to a different question, in this post about how the heads of Europe responded to the execution of the royal family, but I'll respond more specifically to your question here.

The greatest difference comes down to the European relations that these two countries had at the time: France was at war with the rest of Europe BEFORE the execution of the king, while England's martial battles were internal (hence the reason the English Revolution is also referred to nearly as frequently as the English Civil War).

I think it's important to get the timeline very buttoned up here: France was ALREADY at war with Austria and Prussia in 1792, with England joining in shortly thereafter. War-mongering had picked up throughout the Fall of 1791, all of this while Louis was still on the throne. Louis wasn't executed until January of 1793, though he had been imprisoned since 10 August, 1792. The heads of Europe were not fighting for the safety of the French royal family, and they certainly weren't invading to avenge the king-- remember, war had been declared when Louis was still on the throne. So while the heads of Europe post-facto did throw in language about protecting the royal family (again I'd recommend my post from a few days ago on that) it certainly wasn't a war to avenge a fallen monarch.

So why was France and the "rest of Europe" at war before Louis was even dethroned? Well, many reasons, which is why history is the gift that keeps on giving. The main reason I'll touch on is the most relevant to your question, we'll call that "France exporting her Revolution"; this is really the crux of why England warred within the bounds of Great Britain, while France was at war externally.

For the revolutionaries of 1789, the Englightenment ideals of "liberty, equality, fraternity" were not for Frenchmen alone, they applied to all men. This was not idle talk either: revolutionaries such as Mirabeau (in his Memoirs of the Court of Berlin) told the subjects of other "enslaved" nations that they could, and should, rise up to overthrow their subjugators. There was a strong sense from the revolutionaries that they were building a new world-- an infectious spirit that bound them together not just nationally, but across borders. In the talks of potential war with Austria there was the idea that the revolution was going to export freedom, and liberate the peoples of other nations.

"But why was there talk of war to begin with?" Good question, because the wars were not started to export liberty, instead it came down to practical dealings with the émigrés and the nations that harboured them, as well as some squabbling on the borders of France... mixed with a healthy dose of that most powerful war catalyst-- hatred.

Let's deal with the émigrés first. These were the French who fled the revolution, usually well-to-do nobles, or clergy who refused to take the oath to uphold the constitution above their religion, and headed by the king's brothers. These brothers-- first the comte d'Artois (future Charles X) and then the comte de Provence (future Louis XVIII)-- set up a court in exile starting in 1789 (when the comte d'Artois 'noped' the F out of France right after the fall of the Bastille, probably my favorite 'nope'-ing in history). This émigré court was not patiently waiting for the revolution to end, they were very aggressivly trying to end it by the tip of their swords. They first settled in Sardinia with the in-laws, but quickly were kicked out and set up shop with the German princes along the Rhine, whose territories bordered France. This leads us into the "squabbling at the borders of France" piece of the war mongering, as there was a lot of acrimonious anger over things like new export and import duties that the revolutionaries were implementing to make trade more free within France. So as these German princes are harboring the émigrés who are saying to ANY nation that would listen "hey let's destroy the revolution it sucks pls give us money and weapons and your armies", you then have the revolutionaries threatening property rights for any émigré not willing to come back to France, which affected some of these foreign landholders. Add into that bubbling Coke bottle the Mento that is "Rampant Austro-phobia" and you have the explosion of war mongering that erupted in late 1791.

Now it is important to note that there certainly was talk about saving the royal family during the course of the war-- this is exemplified by the (in)famous Brunswick Manifesto, which was issued jointly in July 1792 by the Prussians and Austrians which basically said "Hey losers if you hurt one hair on the royal family's heads we're going to walk in and destroy you", but the war itself was not waged as a honor mission to save the French royal family, especially not to avenge them since, as established, they were still well on the throne when the war broke out.

Finally I'll toss in the context of the time: Europe was hot off the Seven Years War (and the war of American Independence) where no one was happy with each other. There was a lot of internal fighting within Europe, and that already bellicose atmosphere certainly played a role, providing a nice bed of tinder to make sure the sparks lit up.

So to sum up, the heads of Europe didn't fight France because of the execution of Louis, the war was already well underway by the time he was executed; moreover, the other heads of Europe had a lot to lose by the revolutionary ideal of exporting the revolution, of expanding France to her natural borders. The revolutionaries were bellicose in their rhetoric, and the Austrians absolutely rose to the challenge. The context then was entirely different than the English Revolution (English Civil War) where yes, a monarch was executed, but in a dispute that had a far lesser effect on the rest of the European powers. England was not already at war with any other European nations, she didn't aggressivly pursue an ideology of change that should take effect throughout Europe; France on the other hand was already at war, and did intend to take this struggle outside her borders.

I'm certain there will be questions; areas I didn't cover well enough. It's a complex subject as these things usually are, but hopefully I painted a good enough picture to answer your question. Please let me know if there are questions-- gaps in the painting as it were-- and I'd be happy to help fill them in!

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Parting Fun Fact: Louis XVI requested his copy of David Hume's History of England when he was imprisoned after the Insurrection of August 10 so that he could read about Charles's execution, the other King who had been executed by his own people, and whom he felt he could identify with. It's fun to compare the two kings, whose fates were so similar, and yet were executed for SUCH different reasons. On the most basic level, Louis was incredibly indecisive, a willow in the winds that would blow whichever way his ministers/courtiers/people blew him. He had no follow through, no backbone. In this Charles was almost his exact opposite-- Charles was incredibly stubborn, and dug in his heels even when almost everyone was screaming in his ear "Charlie that idea is awful, do NOT bring the Book of Common Prayer into Scottish churches dude!" He was SO inflexible, he forced the opposition into extreme positions at every turn. I like to think that there is some sort of afterlife, and Louis and Charles are floating around Andromeda or something idk and can talk about the irony in their executions for nearly opposite personalities.

My answer is too long for Reddit, so I'm splitting into two comments.

The big difference at issue was scope.

In England during the 1640s, what got everyone riled up was Charles meddling in religious affairs, and some, creative, interpretations of his powers as king. I'll deal with the second one first. So, regarding the powers of the king, it is important to first note that, although England was an absolute monarchy at the time, a big power that Charles lacked was that of taxation. Only Parliament could levy taxes. (It is also important to note that there weren't any fixed term parliaments back in the day. Instead, they only met when called by the king, a function coming from Parliament's evolution out of medieval great councils.) When previous parliaments had met, they had acted as almost a de facto rubber stamp for whatever the king had asked for, although they often took the opportunity to air their grievances regarding the king at the same time. What is important though is that, regardless of their issues with him, parliaments normally gave the king what he asked for. Another tradition regarding parliament was that, when a new king ascended, parliaments traditionally voted him 'Tonnage and Poundage' for life, which were effectively import and export tariffs on all goods at set rates for the life of the king. These were intended to pay for the standard operation of the kingdom, so if parliament was getting called again during the king's reign, something important must be happening.

Everything changed when the Fire Nation attacked. Charles I was of the House of Stuart, and Charles' father King James I had been King of Scotland before ascending to the English throne following the death of Elizabeth I. This presented two issues; the first being that, although having technically converted to Anglicanism upon ascending, the House of Stuart was widely considered (most likely rightly so) to still hold Catholic sympathies, with some going so far as to call them 'crypto-Catholics.' This Catholic leaning didn't sit well with the Anglicans, nor the more radical Protestant sects appearing in England at the time, notably Quakers, Puritans, and in Scotland, Presbyterians. [I don't mean to imply these factions didn't exist prior to the reign of King Charles.] The second issue was that Scotland had been a much more thoroughly absolute monarchy than England, in that taxation for Scotland was the prerogative of the king. When Charles first ascended, Parliament declined to grant him tonnage and poundage for life, rather only granting it for three years. Moreover, they insisted that Charles not only listen to their complaints, but swear to act on them. Charles responded by ending the parliamentary session, and governing 'in personal rule' from 1625 - 1640, meaning during this period no parliaments were called. This raised the issue of finances though, because without taxes or tonnage and poundage, Charles had no source of income to finance royal activities. However, Charles could levy fines and fees without parliamentary approval, the difference of note being that taxes are a regular occurrence each year, whereas fine and fees were understood to be one time events. One fee of note is 'Ship Money,' a medieval doctrine which allowed the king to demand a ship, or the money to purchase a ship, in cases of national emergency. This was permitted because it was meant to be invoked in situations of national emergency, such as an imminent invasion by France, Spain, etc., meaning there would be no time to call a parliament. However, what actually constituted a national emergency was solely within the king's authority to decide. This led Charles to just declare a national emergency and demand the funds, which he also levied from the inland counties. Whether he was taking a more enlightened view of the national interest, as if the coastal counties couldn't repel the invasion, the inland counties would be in trouble, or not, the issue was that it was evident there was no national emergency. Charles was acting within the letter of the law, but not the spirit. Now, the first year it was levied, there was grumbling, but there wasn't too much opposition. The next year though, Charles again declared a national emergency, and again demanded Ship Money. However, this was the second year, which meant it was now a regular thing, which meant it was considered as a tax, which Charles could not legitimately levy. This led a well known former MP John Hampden to publicly refuse payment on this argument. Charles had him arrested, and at trial, Hampden was found guilty. The rationale of the court, which ruled 7 - 5 in favor of Charles, was that regardless of regularity, Ship Money only required a national emergency, which the king could declare by himself. While Charles was technically correct on the law, he was violating the spirit of the law. Charles' actions were technically lawful, yet England at the time already had what is known as an unwritten constitution. Charles did have the authority to act as he did, but he was far from winning any popularity contests amongst his middle and upper class subjects. [I say this primarily because it is far harder to know what the lower classes were thinking, given the lack of available source material as compared to more middle class merchants and upper class landowners]

So, to tell the story properly, I need to do the religious issues here, and then go back to the creative interpretations of things. Charles had a degree of reverence for a single religion within all his territories. I'm going to ignore Ireland here given that they won't enter the Civil War fully until after it begins, but for Scotland, churches there used 'The Book of Common Order,' which, while similar' was different from 'The Book of Common Prayer,' which was the text used in England. Charles ordered that Scotland switch to the Book of Common Prayer, which prompted riots, and then a full military revolt.

This left Charles in a bind. He needed to put down the revolt in Scotland, but, even with everything else he had done, he lacked the funds to raise an army to go to Scotland. This prompted Charles, in 1640, to call the first parliament since 1625. After 15 years of watching the above, in short, they were not thrilled with him, and this sentiment was only exacerbated after Charles demanded they vote him funds and then disband. Several parliamentarians led calls against this, leading Charles to personally enter the House of Commons with a small band of soldiers.

Although the MPs he sought to arrest fled before Charles arrived, entering the House was, symbolically, the straw that broke Charles' back. English monarchs never set foot in the House, let alone meddled in parliamentary affairs. That Charles not only violated parliamentary sovereignty by entering, but also by seeking to arrest its members, led many MPs already inclined against Charles to brand him as a tyrant. At this point the Civil War begins.

I will apologize in advance for how quickly I deal with France. My research regarding this issue has primarily been surrounding England, so my explanation of France will be comparatively brief. Getting into France, the lead up to the Revolution had two primary classes of causes, economic and philosophic. Dealing with the economic first, remember that France in 1789 is only a few years removed from the Seven Years War, in which France lost the majority of its North American colonies, such as Quebec, to the British, but also incurred a massive war debt. France at the time was divided into three estates, the clergy, who were exempt from tax, nobles, who were de facto exempt from tax, and everyone else, who all paid taxes. This war debt fell largely on the Third Estate. Moreover, while France was, like England before it, an absolute monarchy, French kings from time to time called an Estates General, which functioned somewhat like Parliament, except that it was really an advisory board to the king, rather than having any power in its own right as Parliament did. Now, the Estates General were far less common than Parliaments, having met last before 1789 in 1614. However, the more common Council of Nobles had failed to deliver on any meaningful economic reform to deal with France's debt which, combined with a grain shortage stemming from several poor harvests, had left France in a very poor state financially. One of King Louis XVI's ministers convinces him to hold an Estates General to attempt to bypass the nobility's objections to his proposed reforms. The issue with the Estates General though, was that each estate had a single vote on the matter at hand. Between the two of them, the First and Second Estates totaled around 10% of the population, but had two out the three votes. [That the Third Estate was often overruled despite being roughly 90% of the population was a major argument against calling the Estates General] Once the Estates General was called though, many Third Estate representatives, admittedly many of them middle class merchants or minor landowners, were confronted with opposition to any reform by the First and Second Estates, including a proposal to reorganize the Estates to make them more democratic, but, more importantly, to break the hold of the First and Second Estates over national politics. The First and Second Estates were not exactly pleased with the idea of surrendering any power, which prompted an impasse.