There's a few factors contributing to this: Firstly, we are the worst judges of how we sound. We're too close to the instrument (vocal cords) to be any kind of good judge of how it sounds, this is why singers of all styles and experience levels have voice teachers.
Then there is the recording method itself. The mic, the source, they all add their own coloration to the sound. And then the source used for playback adds its own coloration to the sound. There's an audio term: crap in = crap out. This is true of recording and listening. The quality of the mic, the pickup (if you use one), the source (sound card, dac, if you use one, program used...), and then the playback - are you using a crappy PC soundcard through crappier PC speakers? Those things will actually make certain frequencies inaudible, changing the character of your voice.
How much money you invest in recording stuff depends on how serious you are. Like anything, there's budget stuff, and there's insane stuff. But if you could get a decent soundcard, a dac, and either proper speaker monitors (I think even Apple sells some M-Audio Monitors?), or proper headphone amp and decent headphones, you'd go a long way to hearing how things should sound.