That's not quite true!

While they're both very similar, both having to do with remembrances or laments for the dead, an elegy is a poem of serious reflection, usually about death, but a eulogy is a speech typically given at one's funeral, praising them and talking about how they lived and how they'll be missed.

So one is more active (eulogy, being performed at the funeral, for a specific person) and the other is more passive (elegy, a sad poem , could be about specific dead or just the concept of death in general).