I've tried fallout 3 before and it just didn't hook me up. I gave up on it in an hour or so. Fallout 4 leaps and bounds ahead of 3.
I'm not a big fan of shooter games because there's usually virtually no progression and the story is usually dull, which only leaves you shooting through hordes of predictable enemies. Luckily, fallout 4 has an open world with tons of progression / play-style customization options.
Story
I can't say I am fond of the story of the game, but then again, I hardly ever enjoy game stories so I'll just leave this aspect for others to review in depth. All I can say is it's not bad - it just starts out generic to the point where it feels like an intended cliche. That start of the game made me skip most of dialogue later on. On top of that the main objective of the game (don't want to spoil) feels forgotten as the story progresses.
Gameplay
If you look at the bigger picture, you just complete generic missions, build settlements and survive. There are enough obstacles and mechanics to keep you challenged for quite a while in this quest. There's also quite a bit of optional content so chances are you will get way more play time out of this game than most other games out there.
Fallout 4 has some mechanics that would take a wall of text to explain. If you really want to see them all, i suggest you watch a lengthy gameplay video.
Only mechanic I must mention is VATS. I'm not even sure what it stands for, but when use it slows down time and allows you to aim and automatically shoot at enemies (and specific limbs). While this mechanic requires some skill points spent into it to be accurate enough it's definitely fun to play with.
On top of that, modding community is very active, so do give mods a shot.
Character Progression
Anyways, story aside, the thing that keeps me interested is the progression in the game. Fallout 4 is how I progression should be done in most games that have skills or trait progression or even equipment.
First of all, character skill upgrades are actually meaningful, unlike most other games. Each upgrade feels like a measurable improvement on your character, rather than being a tiny bonus that only works in specific circumstances. To give you a few examples: shoot through walls, chance to reflect bullets, chance one hit kill, etc. Naturally there are a bit more boring ones like increasing damage of your Heavy weaponry by up to 100%, but they really play into the game well. The reason I like these skills so much is because game plays completely differently depending on your skill build.
Weapons have randomized stats and bonuses add to the character development and greatly improves the feel of rewards in the game. On your travels you will find loads of different weapons and armors and, truth be told, most of them will be useless. That only brings you to appreciate the good weapons.
Moreover, if you invest a few skill points into crafting, you can modify almost every weapon in the game. Further more, when i say "modify" I mean you can customize almost every part of the weapon (with exceptions on some weapons).
Finally, do you know how games like to add collectibles into the world? The kind of items that are sometimes hard to find, but are completely useless if you're not an achievement hunter. Well guess what! They actually matter in Fallout 4! They give you various permanent bonuses with each one you collect!
Conclusion
Story is not my concern in games so I'm not rating that at all. The gameplay is amazing!
If not for some weird bugs, that I could swear date back to first year of Skyrim, I'd give it a 10/10.