Nikon has no issue with the Canon T3i aka 600D. Why should they ? Thats the precedessor of the precedessor of the current 700D. Thats a camera from 2011. The Canon equivalent to the D3100.
And IMHO D5100 >> D3200. All the D3200 offers is better video and more megapixels. The D5100 has the better sensor and the super practical flipscreen (sadly the D5100 and its successor D5200 are still the ony Nikon DSLRs with such a flipscreen). I would never trade my D5100 for a D3200, ever.
Deciding which company you go for is done this way: you get into a good photography shop and take each camera in the hand. Check out where the functions are, how you like their placement, which camera seems to be more intiutive to you.
Another criteria is of course: what do your friends have. Because you can share lenses (and flashes etc) with each other if you all have the same.