Generally speaking, the difference in tubes is affected mostly by the power amp tubes, since they differ more greatly from amp to amp and are not usually interchangeable. Almost every common amp uses 12AX7s in the preamp though, which can be swapped easily for 12AY7s, 12AT7s, or 12AU7s for different levels of gain.

In the power amp though, the general divide comes between power tubes that are used in American amps and therefore have the "American sound," which are almost always 6L6s or 6V6s, and the tubes in British amps that give the "British sound," like EL34s and EL84s. With American style power-amp tubes, you get more chime, a scooped midrange, and clear articulation. British style power-amp tubes usually have a tighter bottom-end, more pronounced midrange, and are more easily overdriven. Generally, 6V6 and EL84s are used for smaller combos, and 6L6s and EL34s for larger amps and heads.

Most power amp tubes are not interchangeable though, and require matched pairs (I.e., you can't usually have one EL34 and one EL84 unless the amp was specially designed to accommodate that which isn't common). They also require biasing unlike pre-amp tubes, which means you can't just swap them out and have everything sound great; they need a bit of tuning.

Obviously this is just a generalization though. There are American amps that use EL84s, and British amps that use 6L6s, and then there are also the more uncommon tube types like 6550s and KT66s. You can also get totally contrasting sounds out of certain tubes depending on how the amp's tonestack is setup, the type of speaker, whether it's a combo or head/cab, the type of guitar you're using, et cetera, but the British vs. American divide is the easier way to simplify it. American = clean, clear, pristine high-end and warm low-end with scooped mids, and British = warmer high-end, tight bass, punchy mid-range, higher gain.