This is the kind of topic that people can - and have - written entire books on. Unfortunately, it's also not exactly my area of expertise (I'm a classic sort who focuses mainly on the First and the Third) so my answer will be briefer than the topic probably deserves. There are hundreds if not thousands of little differences - many of which have significant social and political implications - but I'm just going to focus on two aspects.
The first, and perhaps most obvious, is the target. The First Crusade was intended for Jerusalem (probably), while the Second and Third were to support an increasingly endangered Crusader States. One thing that became increasingly clear in the wake of the Third Crusade was that Jerusalem was very probably indefensible given the current political status quo, and the idea was formed that the Crusaders should focus on building a bulwark to keep Jerusalem safe before attempting to reconquer the city. With this goal in mind, basically every major Crusade after the Third was launched with the intention of taking Egypt, not the Holy Land.
This might seem strange, but Egypt was extremely wealthy (it was the wealth of Egypt that made it possible for Saladin to carve out his empire and eventually take Jerusalem from the Crusaders). Egypt also had a significant Christian population (they were Coptic, not Catholic, but still), so establishing a new ruling class of Western Christians wasn't complete madness. That said, conquering Egypt is easier said than done. The Fourth Crusade famously went somewhat awry (mild understatement) and sacked Constantinople, but the Fifth and Seventh Crusade actually involved (failed) invasions of Egypt. The Sixth Crusade did focus on Jerusalem - and actually secured the city via peaceful negotiation - but it was highly controversial at the time and the city didn't stay in Christian hands for long. The Sixth Crusade is also sometimes wrapped up with the Fifth Crusade since Frederick II technically took his vows as part of the Fifth Crusade but delayed going for so long that he was basically launching his own mission.
The Eight Crusade was an attempt by Louis IX to take Tunis, which while technically not Egypt does feed into an overall "conquer northeastern Africa" strategy, while the Ninth Crusade actually began as part of the Eight Crusade but Edward I of England arrived too late (Louis IX was already dead) and instead continued on to Acre to make one last effort to defend the Crusader States. It achieved a few victories, but nothing of consequence. Acre held on for another 20 years, before its conquest by the Sultan Baibars in 1291 - after which point Crusading changed a lot.
The second shift was slower and a bit more stop and go, but over the course of the twelfth century Crusades became increasingly associated with singular monarchs. The First Crusade somewhat famously had no kings on it at all, while the Second and Third hard several. The Fourth Crusade and Fifth Crusades were in part an effort by the papacy to reassert their control and exclude major monarchs (the French Monarchy was actually simultaneously distracted by the Albigensian Crusade and so the French were only minor participants in the Fifth Crusade) , but both of them ended in failure and somewhat scuppered that plan. The Sixth Crusade was entirely the project of Emperor Frederick II, while the Seventh and Eighth were projects overseen by (St.) Louis IX of France, and the Ninth was Edward I of England's pet project.
A big part of the reason for this transition was that as Crusade expenses rose, and profits declined (failure is not very profitable), only monarchs could really sustain the expenses and mobilise the knightly class enough to get them done. The Fourth Crusade was infamously led astray due to its overwhelming financial debt to the Venetians. This new system had a major downside, though, which is that monarchs had to eventually return to their own kingdoms - ideally sooner rather than later. This exacerbated a problem that had plagued all the crusades after the First, soldiers came in from Europe, mixed up the status quo, and then left leaving the few settlers who remained with more territory than they could reasonably defend. An increasing trend was to place land in the hands of the military orders (Knights Templar or Hospitallers usually), but those groups were also often short of military members and while they grew wealthy they were increasingly unable to handle all the lands they ostensibly controlled, which created a sort of death spiral for Western Christian territory in the Holy Land.
The two excellent general histories of the Crusades that cover both early and later parts are Thomas Asbridge's The Crusades and Christopher Tyerman's God's War.
Malcolm Barber's history of the Templars, The New Knighthood, is showing its age but his discussion of the problems with monarchy led Crusades and how it fed into the Templar's power is still really interesting.