The two cameras are very close in overall image quality. Community comparisons and test data suggest the D3300 is only slightly ahead overall, and in practice the lens you use and your shooting technique will usually matter more than the body.

The main image-quality difference mentioned is that the D3300 does not use an optical low-pass (anti-aliasing) filter. That can let it resolve a bit more fine detail in some textures and patterns. The tradeoff is the usual one with no AA filter: potentially more risk of moiré in certain subjects.

One answer also notes the D3200 may have slightly better dynamic range, so there isn’t a dramatic across-the-board winner.

Bottom line: the D3200 is not “much worse” than the D3300. If you compare them side by side, you may see a small detail advantage from the D3300 in some scenes, but for most users the difference is minor.