I feel like I should be the foremost authority on all things Fallout (Been following the games since '97 and I've got every title except for the Android game cause my phone is lame), but Whiskey, grey, and ACamp have summed things up pretty well here.
If you enjoy Elder Scrolls games, there's really no reason why you shouldn't enjoy Fallout. In my opinion Fallout is more enjoyable because it doesn't take itself as seriously, meaning there's a lot more humor in the game (Both subtle and apparent). Fallout seems to give you many more options when it comes to endings and ways to complete quests, and I love options, but as someone pointed out not everyone likes to stray from a linear story line. Not everyone enjoys the post-apocalyptic setting, but the reverse is true too. My buddy is a big Fallout fan too, and I tried to introduce him into Elder Scrolls. He played the game one night never to pick it up again. While Fallout 3 and New Vegas didn't have the infinite number of companions that Skyrim had, I liked the fact that the characters were far more robust when it came to dialogue and in the case of New Vegas they had actual quests associated to each of them. Makes it that much more painful when a companion bites the dust on Hardcore mode. I know Fallout fans that will debate to death who the best companions in New Vegas are, or who their favorites are, but no one really cares if Lydia dies in Skyrim. They just recruit a replacement Lydia.
I agree that Fallout has more action than Skyrim, but they're two very different games and both still have plenty of action, it's hard to quantify. I think the difference is that Skyrim has a lot more inhabitable locations with civilized people, while Fallout has a lot of wasteland with roaming mutants and raider types. VATs is a great tool, but the nice thing is you can use it as much as you like or not at all, it's not required. VATs is especially nice for players that aren't big into first person shooters and enjoy more of the RPG element, or for those that prefer to run around in 3rd person mode.
Lastly, I'm not going to go back and look, but someone said that the wasteland mutant genre is played out? Wat?! First off, the fantasy genre has an infinitely greater number of games dedicated to it, which is fine cause I love that genre too. More importantly though, Fallout is pretty much THE game that started the post-apocalyptic, mutant slaying, RPG craze. Every game that has come since is and will forever be borrowing from Fallout. The only game that can really lay a claim to revolutionizing the genre before Fallout is the game Wasteland...and the people that created that game are the creators of Fallout so they're all the same.