Rick's already answered. I'll just add that "Thanks for the dinner and the gifts" is what I'd write if I have plenty of space and ink. If I want to economize I'll drop one of the articles.... it wouldn't matter to me which one. But rather than leave BOTH of them out I'd go and find another pen.

When two more things are considered together, we can often omit the second article or determiner. For example: 'You should wear a shirt and tie'. 'Can you pass the salt and pepper?' He left his money to his son and daughter. In your sentence, 'the dinner and gifts' are the two things which you want to thank the other person for, so you can consider them together and miss out the second 'the'.

A grammar guru might answer as to WHY, but in my experience, they mean the same and are fine. I think it just depends on how formal you want to sound. You could even say "Thanks for dinner and the gifts". I would not SAY (verbal communication) "Thanks for dinner and gifts", though I think it would and could be used as a quick note on a thank you card. The most formal response, in my opinion is to not use contractions and to use the articles. So: Thank you for the dinner and the gifts" to me is most formal.