Though the thread title only mentions similarities, the OP also asks about differences. Let's see if I can distill some of both:
- Both are post-apocalyptic settings after a nuclear war; but each has a different take on it. The nuclear war in Fallout took place in a mid-21st century that had reverted to 1950's styles, with rayguns and shiny future-tech. The nuclear war in Wasteland took place in 1998, so Wasteland has an 80's take on future tech, and is generally more realistically grounded, though with some gonzo elements and a cyberpunk sheen. I think it'd be a mistake to describe it as merely grittier, but it was a more sober and realistic take on nuclear war.
- Interestingly, while Fallout was a post-Soviet Union game, Wasteland was released at at time when the Cold War still had the theoretical possibility of turning nuclear. We could speculate on how that might have influenced the development.
- The robots in Fallout are more like 50's Tom-Swift-style mechanical men. The robots in Wasteland are more 80's Robocop/Terminator-ish, though they come in various forms.
- Fallout 1 is set in California, Wasteland is set in Nevada and Arizona, mostly in the desert.
- Fallout was brown, while Wasteland is more colorful. Partially due to technical limitations, but it does give it a different feel. There are green places, and places the war has barely touched.
- There are more groups who preserved technology and who have set up their own enclaves (the Rangers aren't the sole representatives of civilization). There are more crazy religious cults too.
- In Wasteland there aren't really any vaults in the Fallout sense, but more people survived the war.
- This is more my opinion, but violent confrontation feels a like a more plausible outcome to some of the situations in Wasteland. You're a group of hardened rangers, not a clueless vault dweller. I've seen a player fight his way through the Savage Village and not realize there was another way that could have gone.
For gameplay similarities/differences, I'd list:
- You play with a group of characters, not as a single character.
- Fallout has stats, skills, traits, and perks. Wasteland has stats and skills, but the way they are used is slightly different: any stat, skill, or item can be used in an interaction. So you can use Picklock on that door, or Strength, or the stick of dynamite you've been carrying around. The system wasn't perfect (I always thought I should have been able to use LAW rockets on the doors), and some interactions were missed, but it gave a fairly robust series of possible interactions in every circumstance.
- Combat in Wasteland used the then-standard Wizardry-style combat menus. (You may be familiar with the parallel development in JRPG's like Final Fantasy. But with less graphics and more text.) NPCs could use automatic weapons without hitting you every time, though they would still tend to waste clips on rodents. Combat descriptors like "fine red mist" are often quoted.
- You level up characters by radioing in to Ranger headquarters and getting your promotion.
- In both games, you have a lot of options to try to resolve a given situation, and you can often choose between very different outcomes.
I could go on, but that's a good starting point.