Generally some increased privacy. This is more significant if you use seedbox to download your torrent. The likelihood of you getting a DMCA complaint or anything alike from downloading from a public tracker (tpb,rarbg etc) is higher. Although some seedbox allow for public trackers, most don't for this very reason. Another point is that, when we talk about "private", there's a community behind it. Greater speeds (because everyone has to seed in order to maintain their download/upload ratio), greater quality of movies, tv shows etc. Often times, private trackers don't have open registration, but rather invite only. Therefore, it's a chosen community because whoever you invite, will reflect your standing on the server. If you invite a dumbass who downloads but doesn't seed, then that will reflect on you. Some say they release files faster and in higher quality as compared to public trackers.
Just some knowledge I have on public vs private trackers. Cheers