Which of the two lens to buy for birding?

I'm an amateur bird/wildlife photographer. I presently own a nikon d5300 and have no plans to upgrade to a full frame camera. I already own a 18 55mm lens. But i'm confused which one to buy extra of the two:-
1) 55 300mm af s dx lens of nikon
2) 70 300mm nikon lens.
Please suggest which one to buy and why… Wat are the pros and cons of both.

The 70-300 would tempt me more, personally I'd invest in the 70-200 F2.8 and don't jump to the conclusion that my advice isn't good, it's what I would do, you might be happy with the 70-300.
The 2 things I would do before purchasing anything are, I'd spend some time on You Tube reviews and comparisons just to get a feel of what other photographers prefer, I'd visit Amazon and read some reviews on both from people who have actually bought the different lenses and then i'd visit a camera store and handle both just to give myself the reassurance that I was making the right decision.
Because I use my camera more than ever now, I find that I could use the advantage faster glass gives rather than the extra reach in focal length. It really depends on the type of shot you are after and how serious you are about your photography.
I occasionally experiment with nature and wildlife photography:

My birding friends have always used the Nikkor AF and AF-S 70-300 mm lenses and have so for years.

The advantage of the 55-300 mm lens is that it continues from your 18-55 mm lens field of view if you happen to need to use a focal length of from 55-70 mm (probably a rare situation)

The added cost of the 70-300 mm VR (about $190) may make your choice a little easier.

If you are instead looking at the $175 Nikkor AF 70-300 mm, then get the 55-300 mm. It is a much better lens

The 70-300mm would be better. The shorter the zoom range, the better optically.

If budget permits, get longer though (>300mm), especially for those smaller birds.

With your 1.5 crop factor the 70-300 (105-450mm in FF) will be a great economical choice for birding.
The VR will be very handy when photographing birds perching. Be sure to turn it on when using a tripod or panning.

The 70mm to 300mm would be of more use, especially for shy birds which are a long distance away, and for friendly birds which are quite close.