Allegiance has no affect on combat, nor does it have any affect on the enemy's ability to take your planets (although culture does, which directly influences allegiance). To put it simply, allegiance represents the percent income (credits and resources) that your planet produces. Allegiance below 100% means the planet is producing less than it ideally should, and vice versa for above 100%. Culture boosts allegiance up to 10% (additional +10% for Advent if you research Allure of the Unity), and allegiance drops based on distance from your capital planet in # of jumps and under the influence of enemy culture.

Culture on the other hand, does affect combat. Provided you do the research, culture boosts:
damage output for Vasari
shield mitigation for Advent
antimatter regeneration for TEC

Also, the Advent Deliverance Engine (culture cannon) provides a damage boost in addition to the mitigation boost from culture. While allegiance doesn't have any affect on enemy colonization, after your planet is purged, the culture lingers for some time. If the culture is too high, the enemy cannot colonize, so it provides a time buffer in which you can counter attack. This also applies to you if you attack an enemy planet with high culture. You will be unable to colonize until the culture dissipates.