jkid wrote:What is the difference between まい and ません?
まい is the negative volitional form, so it's the plain opposite of ましょう. So your question is what's the difference between the plain and volitional forms?
私が持つ。
私が持ちましょう。
The plain form simply is what it is, whereas the volitional implies a conscious choice or decision made by the speaker, i.e. his volition. I've heard this a few times in Jpod dialogues when the speaker makes up his mind about something, but as far as I remember, it has only ever been explained (repeatedly) in terms of invitational situations.
As for まい, it's almost always used in written Japanese, although it does pop-up in a few expressions. ません can carry a negative volitional meaning, so that's usually used instead. You can't use まい to invite somebody not to do something, either. For that, you add ~のはやめましょう after the plain form of the verb.