Depends,

After breathing is well coordinated, from the technical point of view it should not matter. At first it may be a very good idea to breath through the nose, even sniffing instead of just inhaling. Breathing through the nose also avoids getting dry, which can happen depending on what you are doing.

But, it should not matter much on the singing itself, for some phrases you can't really just breath just through the nose, and on more complicated passages, it may be a very good idea to keep the posture you are going to use right from the inhale, which will involve keeping the mouth open, etc.

As for the recording thing, there is the aspect that Tommy mentioned, which is something I would consider more as an effect. And there is also the aspect of making it sound more natural, it does not need to be in evidence, but if you take it off or don't breath it can sound a bit artificial, depending on the context. If the producer/engineer feels it is missing a breathing sound, then it's indeed their jobs to fix it, there is no precise science to determine this.

The difference in loudness from the audible inhale to the sung phrase can also help give notion of dynamics and proximity. Since the breathing should be quiet in relation to the voice, if it's very audible, it will give a notion of a very close, probably lower intensity voice.

breathing through the nose is healthier (the hairs filter gunk), and the effect of the inhale has the tendency to be deeper due to a vacuum effect, but as Felipe said, for the most part for singing it won't really matter so much.

I personally inhale in the nose at all times.