Out of breath means just that, that you are physically out of breath.

Breathless is generally used in other phrasing, such as "breathless with anticipation" or a similar "breathtaking candor 惊人的坦言." Usually it's used metaphorically 隐喻地 [,metə'fɔrikəli] to say that something so exceptional has happened that it figuratively 比喻得 took your breath away. Out of breath means you're literally out of breath.

If you are trying to say you are literally physically out of breath from exercise, I think it would be more elegant to use out of breath instead of breathless.