Dom is much more fast-paced. In SR I feel suffocated starting from level 1 and farming for enough gold to start to have fun. By the time things start to get fun you probably already know whether you've won or lost. You can't play a champion in a different role because Warwick is ALWAYS jungler and Ashe is ALWAYS carry and you MUST have a tank. Not to mention, if you're just learning those roles, and you don't even know what a jungler or a carry IS, you probably cost your team the game.
SR feels like Mario Bros 1.
Level 1-1, 1-2, 1-3: Make gold by killing minions. Except it doesn't matter how much damage you do to the minions, only that you get in the last hit. WTF? At least in Mario, you don't have people stealing your last hits.
Level 2-1, 2-2, 2-3: Things get interesting. Your players always have the Fire Flower and skill is actually required. Except that that the whole game was mostly decided in level 1-3 when the enemy carry got more minion corpses hoarded than your carry.
Level 3-1, 3-2, 3-3: The fight is over. You know you're gonna win (or lose) all that matters is finding that a couple of extra points to finish your game with the most style, like bouncing off bad guys for 99 lives.
And the whole game is the same again and again and again. There's a bit of skill involved in the middle when things are actually challenging... but you probably already know whether you can do it or not.
Dominion is more like the Legend of Zelda: Ocarina of Time.
You start with a few moves that will be useful the whole game, but you still have to work to get your superpower (Ocarina/Sword/Shield).
The general goal of the game is the same every time you play it (get three caps long enough to win) but how you get there changes depending on your mood and style.
Even when you've played it many times, you always find something new to surprise you. You may have Guarded the Bot Tower successfully before, but this time despite having the same skill you get overwhelmed. Or last time you lost but this time you got help.
Say you have two teams. Players A, B, C, D, and E comprise one team. Players 1, 2, 3, 4, 5 comprise the other. In Summoner's Rift, you spent 20 minutes farming, you've had a few ganks, and things start snowballing. The rules are always the same: don't killsteal, let the carry get the killing blow if you can, don't farm the carry's minions, communicate MIAs, etc. You probably already know who's winning by level 10. A comeback, if you're losing, is highly unlikely. The next game, you play against the same team, and you all chose the same champs. The specifics of this fight aren't going to change much. The carry is still gonna be in mid. The jungler may change his route, but he'll still be jungling and ganking. First Blood will probably still happen at top, and you'll probably take the tower down in the same order. The only way you can really switch it up, if you have the same players, is to switch champs. And everything is about the carry. Feed the carry, protect the carry, kill the carry.
In Dominion, those same ten players, with those same ten champs, have many many more options. Sure, it's probably the same bot champ. But that is probably the only thing that will remain the same. Everything else will change, and it's more than just the fights that will change (which they will); the strategies are CONSTANTLY changing and evolving in order to win. The rules, such as they are, are much more like suggestions. You can do a lot more with individual skill... by learning Map Awareness, you can figure out yourself where the enemies are, or where they're likely to go. You need to cap three towers... that's the only hard rule. HOW you accomplish that can be by many means: kill enemies in the FoW so they can't cap. Defend your tower. Kill their defender. Take their bot and mid and bottle them inside. Even when victory seems assured, you can't just coast to victory; if you don't keep up your vigilance, the enemy team is likely to come back and counter you. To me, that means you weren't good enough to win because you wasted your lead. They were good enough to endure and survive, props to them! You don't have one single target plus enemies, you have FIVE. And it's not all carry, carry, carry.
Yes, it's chaotic. Glorious, glorious, chaos. You learn faster, you move faster, you think faster, you react faster. You make split-second decisions that may or may not cost you a battle, but one single mistake won't cost you the game... you may lose the battle, but you can still win the war.