The card qualifications to enter these two competitions are the same, but the rewards are different.
All Star competitions: Includes rewards only from the scarcity of the competition - e.g. Limited All Star competitions will have only Limited rewards.
Pro competitions: Includes rewards from the same scarcity and the next scarcity better - e.g. Limited Pro competitions will have Limited and Rare (the next best scarcity) rewards.
Reward percentages: There is a higher percentage of rewards distributed in All Star competitions; for Pro competitions, the reward percentage is lower but you are able to get better scarcities.