What is the difference between “tits” and “boobs”?
From Word Net Search: Boob
Noun
S: (n) dumbbell, dummy, dope, boob, booby, pinhead (an ignorant or foolish person)
S: (n) breast, bosom, knocker, boob, tit, titty (either of two soft fleshy milk-secreting glandular organs on the chest of a woman)
Verb
S: (v) drop the ball, sin, blunder, boob, goof (commit a faux pas or a fault or make a serious mistake) "I blundered during the job interview"
(I was not aware of the verb form, but it follows from the first noun form)
Boob could also be used to refer to a person who is acting like a clown, or it could be used to refer to something that induces cluelessness. We have a nickname for television in the US (although it's going out of fashion) called the boob-tube. I assure you it's not because it shows porn.
Noun
S: (n) breast, bosom, knocker, boob, tit, titty (either of two soft fleshy milk-secreting glandular organs on the chest of a woman)
S: (n) nipple, mammilla, mamilla, pap, teat, tit (the small projection of a mammary gland)
S: (n) titmouse, tit (small insectivorous birds)
(I was aware of titmouse but not a small bird called tit)
Tits is generally the word for mammary glands, being a corruption from teat. That's what @ghoppe was onto with his, where you emphasize the pointy bit of the end of the mammary gland. It's more evident on a dog or pig, where there are a line of teats (or tits). The fact that this usage has spread to also refer to women is likely from the fact that country/urban life has comingled more of late than of years past.
The usage to associate tits with boobs however, is purely sexist. In polite society you should mention neither, referring instead to the bosom that @Robusto mentions, or breast, but only when you must. Bosom indicates "that feeling of your grandmother wrapping her arms around you when you're scared" and breasts is generally seen as more clinical in nature.
Generally in polite society you would just avoid references to a woman's breasts, and let it go.
When breasts are being discussed (such as in the context of bra fitting), women will often use the word "boobs" amongst themselves.
With men and men alone, tits is fine.
NOTE: I edited in response to a comment and decided I would give a little more attribution.
I think there is a slight difference in what you're emphasizing. They both refer to female breasts, but tits emphasizes the nipples (pointy bits) whereas boobs evokes an image of the round parts.
As others have said, both refer to a woman's breasts, but the main difference is actually in the connotation. While both are euphemisms, boobs is a slightly more socially acceptable term. Women, in my presence at least, use the term boobs; tits is more a term that men use and generally in a more lurid fashion.
They're both terms for female breasts. Women tend to refer to these as "boobs" and not "tits"; with men it can go either way, but a good part of the time they'll use "tits". Other (somewhat vulgar and borderline spring-break moronic) terms are knockers, twins, hooters, hoo-hahs, etc. Puritanical people sometimes call them bosoms, which is strange because bosom refers to the upper chest in general and few people have more than one of those.
By the way, to reference a woman's breasts by any term in her presence is coming on pretty strong, and in a business setting it can be tantamount to sexual harrassment.