650D t4i vs D5100
As some other people pointed out, both will allow capture of whatever you're looking at. The camera doesn't have a whole lot to do with them being "amazing" or "great."
A couple of points before the cameras.
Books, not the internet, are the place to start if you're seirous about learning about how to make better images. There's a lot of things to be found here and you'll learn some words to research, but it's usually incomplete or skips a few steps that you're presumed to be able to fill in.
Being good with editing software is probably as important as the camera. I luckily had a 10 year casual relationship with Photoshop and Paintshop so I was able to make sense of Lightroom, but a year and a bit later (and several other plugins) I just feel like I'm getting good at it. It makes all the difference in your final product.
That said, there's no substitute for checking the cameras out in person. When I was deciding on the D5100/T3i I was set on the Canon from online. It was December last year so prices were higher for the new Nikon at the time. I went to the store, restored defaults, took some pictures on a SD card and went home to compare. I didn't like the Canon layout or feel of the body at all. I liked the photos less.
Now it's 7 months or so later and I think I made the right decision. There are a few things that I wish were better about the D5100 - high speed flash sync for one, but I didn't know that was a thing when I was buying it. Another big one is how the self-timer works. It sets the interval between every shot so you have to wait 4 seconds for a 2 second spaced 3 exposure bracket. My LX5 fires off a burst of three all together after 2 seconds, which is great for HDR , but I didn't think to check that when researching DSLRs.
I also didn't realize what a chore video was going to be with it. It produces great videos, but you have to work at it. I am more prone to just grab my LX5 if I need to make videos of the kids. You just put it on auto and push the button and you're good to go. I also don't care much about video quality. It's amusing and good to have as a record, but it's not like I'd subject other people to it.
The other thing about the Nikon is the 35mm f/1.8 lens. I haven't taken mine off in the two months since I got it. For kid photos I don't think you're going to find a better option for quality/budget/focal length.
You also need a flash.
For the difference in cost the D5100 is the clear winner for me. You can pick up some other tools, accessories, and you'll quickly find out you need.
Where I am now I realize that despite doing what I thought was a ton of research I really didn't know enough about photography or cameras to make an informed buying decision (you don't know what you don't know and will not without taking a bunch of pictures and actually using a camera first). That was after spending a lot of time moving up from a random point and shoot to a LX5 and learning about the differences there and shooting manual RAW so I was much better informed than I was before that.
Whatever you choose you'll probably realize at this tier of camera you're lacking a number of features you would really like, but don't even know exist at this point. The important thing is that you get and start learning and try and figure out a long-term plan to get your wife to let you buy more stuff.