I wouldn’t draw such a sharp distinction between the two, as they’re both dramatizations of actual events. I doubt either is totally accurate (and I’m sure I could Google up a few dozen supporting links in a heartbeat), but I think they both sought to achieve a similar level of authenticity.
They make great companion pieces, and I love popping in the Apollo 13 DVD when FTETTM reaches that moment. (The show focuses on the media’s role at that point in the space program’s history. Interesting, but much less so than what was happening with Lovell & Co.)