nuances - Difference between 「～のない～」 and 「～がない~」?

For example, plugging these into Google, I can find stuff like, 自信のない人 and 自信がない人. Or, 愛のない生活 and 愛がない生活. I found a movie called 顔のないスパイ. How is this any different from 顔がないスパイ?

I've asked Japanese people this, and the only answer I've ever gotten is that that's a hard one to explain, and that there really isn't much difference worth noting. I'd like to know the gritty details though, what little nuance each holds.

I feel like I see this difference with "ない" more than anything, but that could just be me. I can't really recall ever hearing the opposite, like 自信のある人. Is this significant? Am I wrong?

Thanks!