The 55-300 is meant only for DX cameras and the 70-300 is a good choice for both DX and FX cameras.
Beyond that, their build quality is noticeably different: the 55-300 is lightweight and the focus and zoom rings are not particularly smooth. The 70-300 is a little heftier with smooth focus and zoom rings. (To be clear, the 70-300 is not built like an f2.8 zoom, but it's nice.)
Image quality: for the price, I find the 55-300 to be quite good. Stopped down a little, you probably can't tell the difference between it and the 70-300. Which is to say, the 70-300 is clearly better wide open.
All considered, the 70-300 is clearly the better lens and in my opinion it's worth the price. If the price of the 70-300 is too high for you, I wouldn't hesitate to get the 55-300; it's a good lens for the price.