I understand that you can still play after you beat the story. Are there any disadvantages to doing this?

There are dragons.

No disadvantages, but its not as good as the rest of the game. You know there is so much to do in the game that you need not even do it at all, just get far enough so there are Dragons in the world and leave it at that.

I guess it can get quite tough with an underleveled character. Everything scales with you, although not so much as it did in Oblivion. But still, having those extra perks can really make a difference. I would suggest doing some side-questing between each storyline mission, just to work on your skills and get some useful perks. You can leave all the "fine tuning" for after the main quest is done.

As the story progresses enemies do get tougher (as well as when you level up), the dragons get way more powerful (not sure about completing the main story, whether they all die or stay) and the story should really be your 'emergency reserve'. Eventually you will run out of stuff to do, and that's when you hit the story as it often opens up more interesting areas and some new misc/side quests. There are the guilds, but their quest lines are like a quarter the length of the ones in Oblivion, and they're all pretty similar.

I guess it really depends on what you want to be in the game or taken out, be it challenging fights or no dragons (I haven't completed the main story, so I don't know what happens to the dragons, although I'm assuming they all die).

I actually think you should finish the main quest first, because the rest of the game is so much better then it