1) They have a single word for "the day after the tomorrow", and another for "the day before yesterday"! Once you've tasted the joys of asatte and ototoi you'll never go back. (Except you will, because people won't understand you otherwise.)
2) Like Latin - and I wish I knew enough about Latin to make the comparison intelligently - Japanese is a very concise language. However, where Latin packs things like mood, aspect, etc. into word endings (or such is my impression), Japanese simply leaves things out if they're obvious from context. For example, supposing I want to tell you I am happy, I might say the Japanese for "Am happy!", or even just "Happy!" If it's obvious from context, there's no need to give that sentence a subject. When it comes to longer sentences, this confuses the hell out of Google Translate.
3) Lastly, though this is a bit perverse perhaps, I rather like the way the word order is so different from English, and the mental gymnastics necessitated thereby. I got a taste of this learning German ("Whenever the literary German dives into a sentence, that is the last you are going to see of him till he emerges on the other side of his Atlantic with his verb in his mouth" - so true), but Japanese takes it to a different level. For example - and not a particularly convoluted one:
"I want to study Japanese with the woman who gave me a book."
"As for me, me-to-book-gave-woman with Japanese study want is the case."
No doubt there are similar pleasures in learning any language, but I've found great solace in starting to wrap my head round this one. (As I said to my mother the other month, "If I'm going to be mildly depressed it's better to get a language out of it than cirrhosis.") Which isn't to say that there aren't also annoying things about Japanese - and perhaps I'll make a separate post about those at some point soon.