While both the UK and Switzerland are generally considered "democracies" and they both have stable governments with low corruption and a very high standard of living, there are actually some quite significant differences between their governments in terms of structure. Both countries provide very nice guides on their official websites explaining how their governments function, which are linked below.
The United Kingdom is still technically a monarchy. Specifically, it is a parliamentary constitutional monarchy. The Queen of the United Kingdom theoretically still has an enormous amount of power, but she hardly ever exercises it. In practice, most of the decisions made by the UK government are made by Parliament in general or the Prime Minister in particular.
Unlike the US, where the President is directly elected by popular vote, in the UK, the Prime Minister is appointed by the Queen from the members of Parliament. The UK is a (really, the) common law system, so precedent has established that the Queen must appoint whomever Parliament votes for, even though this is not actually written into the Constitution of the UK. Parliament consists of two Houses, the House of Commons, which is directly elected by popular vote, and the House of Lords, which is partially elected, partially appointed by the Queen, and partially made up of hereditary offices still inherited all the way down from the feudal system.
As you can see, democracy in the UK is actually quite indirect; the people elect (most) of the Members of Parliament, who then de facto elect the Prime Minister (who is actually technically appointed).
Switzerland is quite the opposite. They have something very close to a direct democracy, wherein all major decisions are made by popular vote in a referendum. (One thing that may make it easier is that the population of Switzerland is only about 8 million people, about the population of Greater London. The UK as a whole has over 60 million people.) There is no single chief executive in Switzerland, but instead a Federal Council of seven elected officials, each of whom serves as Federal President for one year over their collective seven-year term. Then they also have a Parliament, all of the members of which are elected directly by popular vote. Switzerland is a civil law system, so laws must be written down; they can't be established by precedent as they are in the UK. And in order to write them down, either there must be a vote of Parliament for minor laws, or a direct referendum of the popular vote for major laws (Parliament decides what is "major" and sets up an "optional referendum") and amendments to the Constitution of Switzerland (which are always done by mandatory referendum).
Switzerland's two legislative houses are the National Council of 200 people, which is elected by proportional representation so that each party gets a number of seats proportional to the votes they received; and a Council of States of 46 people, which is elected by a plurality vote in each of the cantons, smaller units comparable to states in the US, such that every canton gets two representatives, except for the very smallest cantons, which only get one. This system is quite similar to the US Congress, where the House of Representatives is proportional to population but the Senate has two representatives elected from each state. The chief difference is that the US does not use true proportional representation, but instead allots districts to states based on population and then elects a representative from each district. As discussed in the Switzerland brochure (much to my delight; how often do government brochures explain political science concepts?), proportional representation favors small political parties, while plurality vote favors large political parties. This is likely why the US has only two major political parties and Switzerland has several.
Thus, democracy in Switzerland is much more direct; while they do elect representatives, there are no appointed or inherited offices at the top (obviously they appoint postal carriers, tax collectors, etc.!) and individual voters directly make most policy decisions.
Cantons in Switzerland also have a great deal of autonomy, making their own local laws and policies under a framework provided by the federal government (this is also much like the US system, where laws vary between states). By default, it is assumed that the cantons will have the right to make policy, unless the Constitution says specifically that a given law is under the authority of the federal government.
In the UK, Parliament (and technically the Queen) have unitary authority, so they can essentially dictate absolutely all law in the United Kingdom. They do grant some powers to local regions (particularly Scotland and Northern Ireland), but this is entirely at their discretion and could be removed at any time. By default, power is vested in the Government of the United Kingdom unless that power is specifically relinquished.
Switzerland is a democracy in the strongest sense, while the UK is more like a de facto democracy or quasi-democracy, or, like I said, the more formal term is parliamentary constitutional monarchy.
Yet despite all these differences, policies between the two countries are really not all that different. Even their economies are quite similar, being highly focused on financial services. The powers that the Government of the UK has chosen to devolve to lower governments are even quite similar in practice to the powers that the federal government of Switzerland has chosen not to take from the cantons. They have taken very different paths, but ended up in more or less the same place.