After looking both of these lenses I chose the 70-300 based on the following:
1.) Better build quality -- it feels more substantial and clicks more assuredly when I mount it on my camera than any of the DX lenses I own
2.) Excellent sharpness -- other reviews indicate that the 55-300 gets a little soft past 200mm, from personal experience the 70-300 is very sharp all the way to 300mm
3.) A/M -- in auto-focus mode, you can manually adjust focus without shifting modes if the auto-focus is not where you want it
4.) FX format -- future proofs me for the time when I can spend the money for an FX format camera and provides a little extra reach (100-420) on my DX camera

Besides the appx $200 price difference, the 55-300 lens is strictly for use with digital cameras with DX sensors only. DX cameras have a smaller size digital sensor than an FX that uses a standard 35 mm frame (24 x 36 mm). Because the lens physically cannot 'fill up' the FX frame it is listed only as a DX (and thus there is no 'equivalent' for FX/35mm).

The 70-300 (that I own) is physically a larger lens than the 55-300 and is designed to work either with a DX, FX, or 35mm film format. Thus, while it is listed as a 70-300 for the FX/35mm format, it gives the equivalent performance of a 105-450mm for the DX frame if you're familiar to what things look like through a 35mm/FX camera. The Nikon website comparsion between the lenses will show you the range in minimum/maximum angle with each lens, noting that both the DX and FX/35mm values are given for the 70-300.

Yes, it is. And is compatible. The 70-300 is so commented because it has faster autofocus, less distortion and better build quality if compared to the 55-300.

But you get drawbacks with 70-300: it's heavier and it's not a DX lens. It is originally designed for cameras like D700 and D3, called full frame (their sensors are bigger). On cameras like your D3100 and my D5100, the 70-300 is actually a 105-450. Lenses for full frame bodies work with their focal lenghts multiplied by 1.5 when mounted on DX bodies (like ours).

I have this same lens with my D5000 and the combination of this lens and camera take great pics -- everyone comments on them. I've attached a couple below for you to see.

I highly recommend this lens for the D5000.