The "Ancien Régime" (from around 1300 until 1798) and modern Switzerland do have nothing in common except they're located more or less in the same area.

AR Switzerland's political structure was a mess, it was less of a republic than a cluster of cities and states that put themselves together by various contracts and treaties over 5 centuries. There was a de facto government called "Tagsatzung" but its power was very limited. In fact Tagsatzung translated to English means nothing else than "hearing" and it was exactly that: representatives of the different entities having a conference about domestic and foreign policies. The framework for this were the aforementioned treaties. There were treaties about defense, trade, taxes and pretty much everything else not only between the single states and the confederation, but also between the states of the confederation.

What's interesting from today's perpective is that regions that became part of the confederation by conquest had no say at all and it's citizens were subjects to the local baliff. The different parts of the confederation were absolutely independent in their domestic affairs and evolved different approaches as to how to govern themselves. In some the power lied with the guilds, in others with patrician families.

This system didn't work too good but it allowed every state to cherry pick and make it's own regulations which is why nobody took the effort to change it.

This vehicle worked more or less until 1798 when the French invaded. The invasion itself showed the limitations of the Swiss system because every state of the confederation fought its own battles and they had no coordination of their movements at all. Some put up a fight and some were even victorious fo a short time but after all the overwhelming power of the French conquered Switzerland rather quickly.

The AR wasn't consistent with the ideals of the French revolution and therefore remodelled after the example, thus the "Helvetic Republic" was born. The HR replaced the cluster of treaties with a modern constitution and changed Switzerland from a bunch of states losely held together into a centralistic nation. The problem was that the more powerful cantons lost influence and power which made it highly unpopular and after the French withdrawal in 1802 a civil war broke out between federalists(those who wanted the AR back) and unionists(people who wanted to keep the centralist organisation). The federalists had the upper hand when France intervened. Napoleon wanted to stop the unrest and set up a new constitution, the so called "mediation dossier". The mediation included more rights for the cantons and reorganised them but still kept much of the HR's structure.

This time the constitution proved somewhat stable and Switzerland remained the same until the end of the napoleonic wars in 1815. In the Vienna congress Switzerland's borders were confirmed(they lost some areas like Valtellina and gained others like Fricktal, relative to the 1798 borders). The political powerhouses of Switzerland started to bring back much of the AR as soon as France was defeated. Church and state remained seperate, people remained equal, but tax and mint authority was given back to the cantons. This made the supporters of the Helvetic Republic rather upset and ended in various conflicts.

Long story short: radicals(who didn't like the current state and wanted to renew Switzerland from its roots(hence radical from Latin "radix" = root)) liberals(who liked Switzerland as it was) and conservatives(who liked the AR pre 1798 even more) gave each other hell with various skirmishes, political intrigue and what-have-you.

This ended in the Sonderbund war of 1847. The liberals defeated the conservatives in a swift war. The main goal of the liberal commander Guillaume-Henri Dufour was not only to win the war, but to win it in a manner which gave the losing side as few reasons for retaliations as possible and he did so surprisingly well; the casualties were astonishingly low and looting was almost nonexistant. Sidenote: Dufour later became a founding member.

The war ended in the new constitution of Switzerland from 1848. Switzerland became once again more centralised, and changed from confederation to federation. Since then not much has changed.

TL;DR: Wer hat's erfunden? Die Franzosen!

Sources:

Kreis: Der Weg zur Gegenwart

Spiess: Illustrierte Geschichte der Schweiz

Maissen: Geschichte der Schweiz