It’s no contest, go with the latest discussions.
If you have a decision between which to give most prominence on your platform, always go with the latest discussions.
The latest activity gets clogged with information which is irrelevant to most members. It includes who befriended whom, objects which members liked/disliked, members sending each other messages via their profiles and a detailed list of discussions related to a single thread.
The latest activity might make your community seem more active, but the majority of the category is redundant to the majority of members. It gets ignored and vital home page real estate is wasted.
Latest discussions however provide a simple navigational method of showing which discussions you need to get involved. It’s those active discussions which drive communities. Contributing in a discussion is far more valuable than any other form of activity. It’s these discussions which you most want to encourage.
It’s a no-brainer, showcase the latest discussions (at least above latest activity).