These two words have similar meanings, however in certain contexts only one of the two can be used or is used most frequently (fits better idiomatically). Can you give examples of such contexts and if possible explain the difference?
Not to be argumentative, but I believe Mr Beckett is technically incorrect. In math, a "proposition" is a theorem, that is, something to be proved. An axiom is something assumed to be true without proof. (To say "an axiom assumed to be true" is redundant, like "ATM machine". By definition an axiom is assumed to be true.) I do agree with Martin about the definition of a proposal.
As Mr Becket is apparently too delicate to explain, in dating a proposal is an offer of marriage, while a proposition is a request for sexual favors without benefit of marriage. If you tell your girlfriend that you want to offer her a proposal when you meant that you want to make a proposition, you may find yourself committed beyond what you were prepared for. If you offer a proposition when she was expecting a proposal, you may find yourself beaten senseless.
proposal is a well developed plan intended for a formal offering, like that for marriage, and it is generally expected to be approved, or, let's say its degree of expectancy as to be approved is above average, while proposition is conjectural and hypothetical and may be rejected or approved. Proposition's weight is far lighter than proposal.