Divergence

One of the most common misconceptions about the Fallout world is that it is just the future of our own world and that it works just like ours. Another misinterpretation is that the Fallout world is the result of a nuclear war with Russia that occurred in the 1950s of our timeline. Both of these ideas are entirely false, partly due to game artwork.

The Fallout world is historically divergent from our own and also is fundamentally different in terms of how science works. The base concept for the setting is 1950s World of Tomorrow after the bomb. This means that before the war, the Fallout world was more or less what the people of the 1950s thought things would be like in 2077. This means housecleaning robots and laser guns were the norm, and cars looked like they were from the 1950s, only with nuclear fusion engines. Nuclear weapons also differed hugely from our universe. Instead of leveling everything in the immediate area upon impact and leaving a large smoking crater, nuclear bombs in the Fallout universe functioned more like conventional explosives in most cases, destroying parts of buildings and blocking off streets with rubble, with the addition of radiation. Washington, DC's mostly intact state is the best example of the difference between nuclear weapons in the Fallout universe and our universe. City design also differed. Washington, DC, for example, looks similar in terms of placement and design, but has some noticeable changes. The National Mall is smaller and narrower, the skyscrapers that defined Arlington in our reality do not exist here, the National Archives is located farther from the Mall than in our reality, museums such as the Air and Space do not exist, and busts of an unknown person are located on many buildings. One of the big differences is that in the Fallout world, miniaturization of computers never occured and the transistor was never invented. Instead, humanity invented fusion power. This is why the computers in Fallout are all of the old reel to reel and vacuum tube type and are generally very large and bulky. Even if these computers are very advanced, the technology to make them smaller never existed. Don't ask about the Pip-Boy, though. They're probably just really small vacuum tubes. The exact historical details of the divergence and even the exact moment when it occurred are unknown and irrelevant, but it is known that it happened at some point after World War 2. All that's important is that this alternate universe eventually spawned the current Fallout world.

Fusion power allowed the Fallout world to do really neat things. A clean, renewable, plentiful and portable source of power meant that things like power armor and energy weapons could be built, as well as all the housekeeping robots.

Arms & Equipment
All three Fallout games use a combination of fictional weapons, and weapons familiar or identical to real world counterparts. The advantage with using fictional weapons, beyond simply respecting the timeline divergence, is that it allows designers to create the weapon they need to fit the game.

Given that the historical divergence occurred soon after World War II, it is possible that war era weapons would exist in Fallout. That said, given that a weapon manufactured in 1941 would be 220 years old by the time Fallout began, finding a functioning wartime weapon is unlikely, unless wartime-designed weapon was manufactured after Great War due to simplicity of production as with Tommy Gun.

Physics in a Different World
The Fallout world doesn't only diverge historically. The universe itself is fundamentally different. The World of Tomorrow theme doesn't only apply to what technologies exist and how history unfolded. It also applies to the laws of physics. In our world, we know that exposure to radiation just kills you. In the Fallout world, however, severe radiation isn't always fatal, and it occasionally produces unlikely or impossible mutations including increased size and in the case of ghouls, extremely long life coupled with a decaying body. Remember classic movies like Them or The Fifty Foot Woman, where freak nuclear accidents caused giant ants or people to appear? That's how it works in Fallout. All science behaves the way it was popularly thought to behave in the 1950s. Similarly, this means that there was no nuclear winter in the Fallout universe since that theory was put forward in our 1970s and only concretized by Carl Sagan and a team of researchers in our 1983. Whenever considering science in Fallout, remember this rule: Science!, not science.