Fallout is one of my favorite game franchises. The retro futurism style of technology that is used is most definitely the charm since the music matches it, but I like having to make it across the nuclear wasteland looking for loot and stuff. I liked Skyrim, but I would prefer something more modern with guns and mutated creatures.
Bioshock is quite similar in terms of the time period. It has the retro early 1900s vibe to it with lots of technology and stuff that doesn't match that era. Bioshock is set in a city beneath the ocean, so you don't have the same level of freedom, but you still get to roam around the city. There is a mod for Fallout 4 that adds the city from Bioshock to the game, so this is definitely a good game to play if you are looking for games like Fallout.
The Resistance Fall of Man games are a lot like Fallout too. They are linear FPS games, so you don't have the open world side of things, but you have a game set in a similar time period with cool futuristic weapons and a good story. The second game was my favourite of the 3 that they released
The Metro games are a little more grim than Fallout and dont have the retro charm, but they are games set on earth after nukes have wiped out most of the planet. You are left to survive in the subway tunnels, making occasional trips outside during the day. There is a nice mix of enemies between monsters, mutants and humans who just can't get along.
Wasteland 2: Director's Cut
Some of the devs that worked on this game were part of the team that created the original Fallout games. Wasteland is a lot more like the first 2 games than the later ones. It is still set in an apocalyptic wasteland, so that alone makes it similar. If you liked the first 2 games, you should get some enjoyment out of it.
A more modern game that looks like it has been heavily influenced by the original 2 Fallout games. It has been well recieved on Steam too. Another game that is worth checking out if you are looking for games like Fallout.
S.T.A.L.K.E.R.: Call of Pripyat
This game is set in Chernobyl, but with a sci-fi monster twist to it. Shares a lot of similarities with Fallout considering Chernobyl was the site of a nuclear melt down many years ago. It doesn't have the post apocalyptic vibe that Fallout has, but it is in the same ballpark