I would say Latin. It is much easier than ancient Greek, both in terms of grammar and vocabulary. Also, ancient Greek is made up of a set of dialects, so if you learn one, texts written in another dialect will still give you trouble. Latin, on the other hand, is a lot more standardized. 

Finally, Latin was used over a much longer period of time than ancient Greek. In the 1900s, for example, German students still wrote essays in Latin. So if you want to look past the classical period and get into medieval history or early science, for example, most writing are still in Latin. 

Latin was my first foreign language at school and Ancient Greek my third. I liked Ancient Greek more, but if I had to choose between the two, I would go for Latin. It's extremely useful as a base for European languages. Ancient Greek is useful too, but when learning languages my knowledge of Latin helped me a little bit more. Apart from that, it depends on why you want to learn those languages in the first place. If you want to study theology or philosophy Ancient Greek is very useful of course, but you may also need Latin.

Both are beautiful languages. I studied Latin 4 hours a week during 8 years in secondary school (plus French and German starting from my native Italian ) and Greek during 4 years. I passionately loved Latin for his logic, his grammar. Never studied too much vocabulary but still had good grades since I considered translation exercises as a mysterious riddle I had to solve. I still miss this playful aspect in any other language I am learning. It was a little bit like solving math problems, playing chess or having a test in logics to measure your I.Q.

Greek was exotic. We were few students to learn it in High School and teachers used to make us feel like we were part of an elite. I love the culture of Ancient Greece. 

But these are just my experiences based on a curriculum that was imposed to me. It depends on what your aim is. If you study philosophy both languages can be useful but not absolutely necessary. If you are interested in roman languages, Latin is the mother of them. If you study theology Ancient Greek (the Koiné version)  Latin ...and Hebrew are almost a must.

By the way, Latin and Ancient Greek are dead languages, in Jerusalem there is an institute called Polis where they teach you TO SPEAK this two languages. Amazing! 

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