I wouldn't - my orthodontist told me not to even use a whitening toothpaste while I had the braces on. As you mention, you will have spots after the braces come off, and there's no guarantee that you'll be able to get them to match up with the rest of the tooth. The chemicals used to whiten teeth can be harsh, and you don't want to subject your teeth to them more often than necessary.
If you're brushing as much as your ortho recommends (mine gave me a time to use), then you may notice your teeth looking better anyway. Also, be sure to cut out staining drinks like coffee, tea, and sodas.
We are all our own worst critics - chances are your teeth only look bad to you, and other people haven't even noticed. TV and magazines have conditioned us to believe that teeth should be snowy white - but that's just not natural or normal.