Yes, I use both. 'Not my cup of tea' is more old-fashioned, but it's by no means outdated, and is definitely still in current use. It has to do with taste. For example, if were asked what I thought of someone's newly decorated lime green and sugar pink living-room, I might say 'It's not my cup of tea', meaning it's not a style that appeals to me. 'Not my thing' is a more modern expression, and it has a wider use. It can refer to taste, or to an interest in general. A 'thing' can be anything that is associated with a person - something that you do or enjoy. For example, you can say 'Animal rights is John's thing', meaning it's a hobby, a passion, or something that is important to him and he identifies with. 'Not my thing' is the opposite of this.

According to Ngram, popularity of “not my cup of tea” has been on the rise since 1820, “not my thing” became popular in 1960 and rises ever since, less than “not my cup of tea”, though. I come across either one in daily life, and I use them in different situation, outlined very well by Su.Ki.

I think the responses pointed out above are correct (taste, old-timey, etc), but I don't think anyone would ever say you're wrong for using one or the other. I would say "not my cup of tea" if I'm looking to sound cute/funny.

From my point of view (northeast US, in my 30s), the "cup of tea" one does sound dated, but maybe it's more common in the UK. I usually say "not my thing".

They are both the same, ' not my cup of tea ' is an older expression, but I still here it from time to time. Not my thing is quite commong. But being a child of the 60s and 70s, I normally say ' it's not my scene '