There are a few differences between the 70-300 and the 55-300. The most evident difference between the two would be the range, the 70-300 is 15mm longer at the wide end than the wider 55-300. Another evident difference is the price, while the 55-300 retails for around $400, while the 70-300 is going to cost about another $200. So, what do you get for another $200? Well, the 70-300 is optimized for FX cameras, but works just fine on DX. In fact, it will give you extra range on DX cameras because of the sensor crop, giving you a 105-450mm equivalent. If you do ever upgrade to full frame in the future (even though you mentioned that you don't plan to in your message, it's still something to consider) the 70-300 will allow you to continue to use your telephoto zoom lens while the 55-300, although it would mount, it wouldn't perform very well.
The 70-300 also offers a few more features that the 55-300 doesn't offer, such as a larger, easier to use focus ring with manual override (meaning that to manually adjust the focus you don't need to switch to Manual mode, you just rotate the focus ring), a focusing window, a much better construction plus internal focusing. I believe that the front lens element rotates on the 55-300mm when it focuses, making it difficult to use filters (but correct me if I'm wrong).
I personally believe that the main advantage of the 70-300 would be that it's just an overall sharper lens, it's a better performer. Sure, the 55-300 is still a nice, sharp DX telephoto that gets the job done, the 70-300 just performs better and is very sharp, the 70-300 is kind of a hidden Nikon gem, it's a very sharp lens for an excellent price.
I personally think that the extra sharpness of the 70-300 is worth the extra $200 alone, plus you get some other features that I mentioned earlier that you won't find on the 55-300. However, it's your call. The 55-300 is still a great lens if you're on a tighter budget, but if you have the extra money, definitely go for the 70-300.
Gear: 7 Nikon Nikkor AI-S and AF-S lenses, SB-700 flash, Nikon D7000, Nikon FM, variety of accessories
"There are no rules for good photographs, there are only good photographs."
- Ansel Adams