8 weeks ago I was aking the same question, and concluded that FLAC had a slight edge over APE. I did novice-level research and it appears that FLAC has a more thoroughly-verified "lossless" functionalitiy. As a novice, I was more assured by the open-source development of FLAC, and my impression that FLAC is a faster-growing standard. This is nothing against APE.
Apparently Matt and/or jriver developed the APE format. This may or may not explain why they refuse to support FLAC. But lack of support doesn't mean any lack of functionality, currently. In fact, I've had one or two instances where the FLAC encoder was less problematic than the APE.
In the future, if Alex B and others can convince jriver, there may be some real advantages to APE. I'm talking about CUE file and track-specific EQ settings imbedded into APE file tags. That at least has been one avenue of discussion, although I'd like to see that functionality within the program, available to all formats.
5,000 FLAC conversions later and I haven't had any problems.