Well, in Skyrim you have a "class" (i.e. a differentiated character) only for a few hours, sooner or later everyone can do everything. In the original Fallout games and New Vegas, the character you create has a lot more impact on the way you play, as character attributes will be altered only a little over the entirety of the game, and skills should be chosen accordingly. Your bookish stealthy scientist guy will not be on par with the grunting melee brute in close combat even at the end of the game. Your in-game character therefore feels a lot more like a distinctive person whose role you are playing. Additionally, Fallout 1,2, and NV also come with optional character "traits" - special modifiers with both positive and negative consequences for you, which are persistent over the whole game, to make your character even more unique. You can for example be more agile at the expense of becoming more squishy, i.e., higher risk of receiving crippled limbs for +1 Agility, or you can deal more damage but be less persuasive and charismatic.
And then, there are the perks you can unlock every two levels based on your attributes and skills, many of which also bend the rules of the game a lot and change what your character can do - sometimes they give you new dialogue options. The last point goes for all the attributes and skills as well, not only persuasion is used in dialogue checks. The game reacts to so many things you do with you character in different ways.