There is a difference in the maximum amount of time you can continuously record with both.
For the 600D/T3i:
"*The file system limits the file size of a single movie clip to 4 GB. If the movie clip reaches 4 GB, the shooting will stop automatically. This translates to around 11 min. for 1920 x 1080 and 1280 x 720, and around 46 min. for 640 x 480.
* Even if the file size of a single movie clip is within 4 GB, movie shooting will automatically terminate when the movie time reaches 29 min. 59 sec."
For the Nikon D3200:
"Movie Maximum recording time 20 min." http/www.nikonusa.com/en/Nikon-Products/Product/Digital-SLR-Cameras/25492/D3200.html
A comparison between the two: http/snapsort.com/compare/Canon-T3i-vs-Nikon-D3200
As far as I can tell from what I have read they are both good cameras. Differences that may influence your decision are the differences in recording time, the articulating LCD screen on the T3i, and image quality. Some user reviews say the Nikon has better quality video. Maybe someone with experience with either will provide more assistance than I can. I have a T4i and have never used a Nikon.