"Teats" are nipples, which may explain some of the confusion. This would also be a vulgar, or at least inappropriate, word if applied to a woman (which is unlikely) but is fine for animals. I have never heard "tits" used for anything but the whole package, so to speak, and I'm not sure where Mr. Wordy hails from but it's certainly vulgar in the US. "TIt" is also used for a single breast.
I could tell the stray dog was either pregnant or a nursing mother by her swollen teats.
Back in my school days, a fan spray-painted the phrase "I'd give my left tit for Vance DeGeneres" on a wall near campus. [True story!]
"Teeth and tits, everybody!" exclaimed the assistant director, as the chorus prepared to go onstage for their big final number. [another true story]