Should I get the Ipad 4 or Galaxy Note 10.1 ?

Well, it's personal preference.
If you are an Apple's fans, then go right ahead and get iPad 4.
If you are an Android's fans, then go right ahead and get Galaxy Note 10.1.

I mean you are comparing an Apples and Oranges.
Battery life, I think iPad 4 might win because iPads are known to have a great battery life.
Performanace, I think Samsung Galaxy is better it has better specs compare to iPads.
Apps, it's a bit tricky. Androids has a lot of developer making apps for educations, games and other cool ass apps while iPads is vice versa, but they do still have a lot of apps just not enough compare to Androids.

Not to mention, 10.1" or 7.8" size? Lol, won't it be better to compare it to 7" to 7.8"?
I mean I think Nexus 7 is great or Nexus 10 decent.

There's no right or wrong just pick one that is suitable for your needs if you have a budget for it.