Full frame lens on a dx body ok?

I just bought a Nikon - 70-300mm f/4.5-5.6G ED-IF AF-S VR Telephoto Lens
without realizing (foolishly) that it's made for an FX camera body and all I have is a DX body (D3200).
Took it out for a test last night and I like it! A lot!
However, would the 55-300mm for DX be a better choice? It's much cheaper. Would the photo quality be any different? Is the more expensive lens a better choice overall because it has better glass or a bigger filter or what?
"Wow me" with your camera knowledge because I have less then a year at this photographic habit and I'm still learning (everyday!).

FX lenses work fine on DX bdies. It is an AF-S lens so it will also autofocus fine. I use one on my DX D7000 body and it is a really nice lens.No, the 55-300 would not be a better choice. The 70-300 VR is a great lens and focuses very quickly. It has high quality glass. The 55-300 is far slower at focusing and nowhere near as good glass quality. Stick with the 70-300.It was a good choice.

Yeah, what Martin said.

Good choice - stick with it. No problem using FX glass on DX bodies.

The 55-300mm quality will be worse.