This shows the differences between the two in an easy to see format.
http://www.dpreview.com/products/compare/side-by-side?products=nikon_d5200&products=nikon_d5300
The D5300 is going to be a better camera.
- It has the newer Nikon sensor (Expeed 4 vs Expeed 3)
- Higher native ISO limit (if shooting indoors in low light - adds more noise to picture though)
- Larger back screen (3.2" vs 3")
- Built in GPS
- Weighs slightly less
Doing some additional research on the difference between the two sensors, it looks like Nikon removed some filters in D5300 which allow for sharper images.
Personally, I'd get the 5300. It's a newer version (although there's now the D5500 so it's not the newest in the entry lineup) that has more features and advancements.
If your wife starts to enjoy photography, it'll "last a little longer" before she wants to upgrade to something else than the D5200.