Typically people use the 70-200 for sports photography however its a heavier lens than the 70-300. But i do have to agree with thefantasticg, my entry level and your mid level camera handle noise very badly when compared to higher end cameras. I was amazed how well the higher models handle noise at 64k even.
I disagree a little here. Pro fullframe cameras handle noise better for sure, but a lot of sports shooters use crop sensor cameras. The weakness with the d5100 isn't in my opinion noise but the fact that it hasn't the most advanced focus system or biggest buffer. I would not however write it off either. Much has been done with less.
OP- you dont specify other than brand the specific model of lens. I owned the Tamron 70-300 vc and liked it but it is not a fast aperture lens. Probably any 70-200 f2.8 will be better. I recently had (on loan) the older Tamron 70-200. This lens is not known for fast focussing yet I found it very good. It also focussed in a church where 5000 iso gave me 1/20th of a second exposure, so it was dark. The newer version is supposedly streets ahead