They’re just labels, and most legislative systems have been influenced by a myriad of sources. However, the labels do loosely point to differences. British and Roman customs are broadly the two most prevalent customs, probably, but the latter definitely influenced the former. There are some tendencies to point out. Parliaments in Europe, despite the French-ish name, are likely to be distantly descended from Germanic tribal meetings that eventually evolved into more formal government institutions. Besides the UK Parliament, the most obvious example appears to the Icelandic one (Althing) – these two have a fair degree of continuity because these countries never really went through violent political restructurings the way many continental powers did. However, the Germanic legacy of tribal moots, which either evolved into or were rejected in favor of (however you prefer to look at it) parliaments in various ways, lasted well into the modern period in nearly every western European country – just as kingship did. Today, whether you have continuity going back to the Middle Ages or not, just about every example of a legislative system in the “free world” has been influenced by liberal ideas of direct democracy, equality under the law, due process, etc..
Political restructurings, even revolutions, often try to conjure older, seemingly eternal, meanings – as if there’s some legitimacy in “getting back to our roots.” In Europe, and the U.S., this often meant the popular Germanic legacies get rejected in favor of what is seen as the height of civilization: the Roman constitution, or bastardized visions thereof. The United States’ selection of a “congress” instead of a “parliament” was clearly a deliberate rejection of the Anglo-Saxon parliamentary legacy in favor of something seen as older, purer, and more romantic.* More specifically, the inclusion of a Senate upon “Capitol Hill”† was almost certainly intended to conjure up images of venerable provincial leaders civilly debating legislation. The House of Representatives is intended to be analogous to Roman legislative assemblies, which were supposed to represent the people as opposed to the aristocracy (I actually don’t know much about Roman constitution, but it’s a safe assumption that it rarely worked out that way). Despite some promising reforms, this shows you that the U.S. political system was really anything but revolutionary – it was evolutionary, but in some ways retrograde.
* Nothing new about this. Charlemagne tried to portray himself as a continuance of the Roman imperial customs. So did Napoleon, and so did Mussolini. And until Napoleon destroyed it, so did much of what is today Germany. Remember the “Holy Roman Empire” (the joke: neither holy, Roman, nor an empire)? And, then look at some other titles: the German Empire of World War I was ruled by a kaiser, and the Russian Empire was ruled by a czar. Note the similarity to the Roman “Caesar.”
† Analogous to the Capitoline Hill, where the Roman Senate met, which you can till visit today