I wear mostly sterling silver, and also collect sterling household stuff, spoons, candlesticks, etc., many vintage or antique as well as new. I've noticed that different pieces can tarnish at far different speeds than others, who never seem to tarnish. Tarnishing is caused by many external influences. It's really a reaction to sulphur in the environment. Even wearing and storing silver near eggs/coming into contact with eggs and onions, for example, can hasten the tarnishing. I also believe that the quality of the silver and the amount of base metal in it may not be the purity stated (I think that not all sterling is exactly the purity stated, i.e., .925 may not be exactly 92.5 % pure silver; there may be either a miniscule amount more or less in the piece.) That also may be why some pieces tarnish much faster, even though "living" in exactly the same circumstances as another piece. This may be why your bracelet is not tarnished, and the necklace is--Tiffany's may have gotten the silver from different manufacturers. Also, some sterling nowadays is laquered with some form of anti-tarnish stuff.
As someone else said, perhaps the skin around your neck and chest is having a reaction. Forgive me saying this, but has it been very hot when you were wearing the necklace, and you could've been sweating more than normal? I don't exactly pour off buckets of sweat myself, but I have noticed that, when it's hot, some heavy sterling pendants irritate my chest and neck, as if they are rubbing on it, and cause a slight rash, so perhaps this could also cause a chemical reaction with your silver? I know that, if I wear non-silver/gold chains, I get a contact dermatitis rash around my neck; or a ring will blacken my finger, yet I can wear the same type of stuff as a bracelet with no itching or rash, etc., so I think chest/neck skin is different.
I would not attempt to clean it yourself, especially with that blue heart pendant being what it is. It may damage that; and, also, being Tiffany's, they should provide good customer service, and replace, or at least professionally clean the necklace for you. No silver, if cared for properly, should tarnnish that fast; in fact, I've NEVER seen silver, even cheap stuff, tarnish that soon!
Let us know what happened!