Think about it simply.
Angle is a measurement in relation to the horizontal.
Rate is a measurement in relation to time.
Therefore:
Best angle will give you the greatest vertical height increase for the shortest horizontal distance travelled.
Best rate will give you the greatest vertical height increase for time airborne.
The aircraft are climbing at different airspeeds, and thus different groundspeeds.
You may be confusing yourself by stating that two identical planes are operating at the same power... don't forget that you could be at 100% power and pointing straight up, completely straight and level, or straight down- or anywhere in between, and so your rates and angles are all going to be different.
It all has to do with your power/thrust curves and which speed you get the max excess.