Multiple manoeuvre penalty

[Qaianna]

This might answer why some unarmed attacks have the Trip trait, especially if they're Agile. Agile weapons use a lower multiple attack penalty, -4/-8. Trips (and other Athletics moves) suffer from said penalty. If you're using an Agile Trip weapon, does using a trip for your second attack (questionable as it may be as a strategy) take the -4 or -5?

[Cordell Kintner]

You never use the Agile trait when making maneuvers. The reason these attacks have the Trip trait is so you can use your item bonus on the roll, just like if you were using an Agile weapon to trip with, like a Sickle.

[Gary Bush]

When you make a combat maneuver, you are using your Athletic skill, not the weapon attacked. The Athletic skill does not have the Agile trait so you would use the full MAP (-5, -10).

As Cordell stated, having the specific trait, like Trip, on a weapon allows you to use the weapon to Trip instead of a free hand. In both cases, you will be using your Athletic skill and MAP, if any.