What is basically comes down to is, if you want to Combo, 1H works with the most stuff. If you want to do most of it with 1 weapon, a Greatsword, or one of the fast 2 handers are extremely powerful. I very rarely use any Hammers, and the only 2 that get much milage are the Longhammer, and Volendrung, because they're slightly faster than Greatswords. Also, Ahtar's Headsman's Axe is another fast swinger, which works with Limbsplitter perks to increase the DPS. That, or Rahd's hammer are unenchanted, so you can use Elemental Fury to further accellerate them, or add Enchantments to make them more effective. Volendrung absorbs Stamina, a lot of it, so you can keep Power Attacking, or Bash&Smash with regular attacks as long as you can keep it charged. And finally, the Drainblood Battleaxe is only availible by running most of the College of Winterhold Arc, and swings at regular BA speed, but drains Heath for more damage, healing you, and weighs less than most Swords, so you don't use as much Stamina to make up for it's slow swing. With Sweep, you can spread Absorbtion (Or some other Enchantment) effects across multiple foes, as well as the Bleed from Limbsplitter. The Battleaxe of Firey Souls has a unique enchantment combining Soul Trap with 10 points of Burn, which you can also spread around if you know how to Sweep.
Flame, or Frost Cloak is a great effect to combine with 2H, because it hurts everyone around you, and lasts a minute, so you don't have to recast it as often. Since the swinging speed is so slow, this really brings up your DPS. It's also an effective weenie whacker, because when you get jumped by 3 Skeevers, Wolves, of Skeletons, it kills all of them faster than you can swing 3 times, or set up for a Sweep. Summons also have Durations, and you can cast them in the other hand before drawing your masher, and stepping into the fray. Flame Cloak, Dremora Lord, and a good Greatsword is plenty of Power.
Of course, you can do all of this (Except for Sweep) with a 1Handed weapon, and more, but another major advantage is Reach, especially if you can learn the timing to use it. This is challenging, because the slow striking speed, even with the fast swingers, require you to start long before they cloas,e and often maneuver mid-swing so you hit them before they get close enough to hit you. A good thing it it's a lot easier to actually land the backward PA for the paralysing perk, even if they don't walk into it, while you'll often come up short with a dagger, or war-axe.
A major disadvantage is the weight of the weapon, which slows you down, and requires a lot more Stamina per Power Attack. This is a major reason why I prefer greatswords (Not to mention the fact that I hate the look of the big blade in the middle of my back with the handle sticking 3 feet over my head with Axes, Greatswords just look more convenient to carry around.) Even with them, I generally use a Bash to stagger, and a regular attack in the opening so my Stamina can recover. A Power Attack uses a lot more Stamina, so while it does more damage, and staggers as well, you might get 1, or 2 of these in, then have to wait to recover. I can get 4-5 swipes in, with another 4 bashes between, which is slower, but does more damage over-all, especially if I have a Cloak doing damage the whole time. Since they're stunned the whole time, I'm in no danger, and they end up just as dead, however, this generally works best in a Duel instead of against multiple foes.
When you're outnumbered, you can power through, or use Hit&Run Tactics so you don't get surrounded. Or, you can armor up, and tank, which is the popular tactic. This is simple, it works, but it's generally quicker to do it with Dual Wield if you're not going to Block. Once you do learn these tactics, they still work with high Armor (And conditioned, or Unhindered really help) I'd say that 1H actually has the Edge as far as usability, but that isn't to say 2H is useless. However, the striking speed makes for a somewhat steeper learning curve, unless you just tank up, and rely on their inherent Damage per Strike. I love both of them, however I mostly focused on @H because the OP was coming from Experience with 1, and transitioning to 2.
Speaking of which, if you get the Ebony Blade, it uses mostly 1 handed Perks for damage, but trains 2H if you want to train both (For an 80 perk build, for instance.) It also drains up to 30 Health, which makes the training a lot easier to survive while you dial in the fighting style, and is the fastest swinger that takes both hands. It doesn't work with the Sweep Perk (Unless they patched it since the last time I bothered to get that damned thing) and there's some other minor glitches that may, or may not be a problem for you. I hate it, not only because of the skill defliction (I actually revel in the irony, and meta-allusion to the aspect of Mephala) but because it's basically a Sword in terms of Damage, but you can't Dual Wield it, and the Enchantment largely overshadows the weapon's damage. It's an all, or nothing weapon which makes it impossible to use with an all-or nothing build. Downright insidious, if you ask me.