So, regular Mission Breakout is the "story" of Rocket breaking out the Guardians from the collection. There are 6 different classic songs with 6 different rise/fall sequences. There are also multiple video sequences but some are repeated in different songs. It's totally random which one you get.
Monsters After Dark takes place after they have broken out and Groot is left behind with all of the monsters from the collection who were released when the Guardians escaped, so now Rocket needs your help to save him. There is one song, which is written specifically for the ride, and one ride profile and one video set. I think it's been reported it's the same profile as one of the classic songs, but you really can't tell. Besides the ride itself, the CMs are wearing emergency vests over their uniforms and are playing along with the escape storyline. The lights inside are emergency red. The preshow video and Rocket animatronic are different to go with the different story. Inside the next area there are different lights and shadow projections of escaped monsters. The outside of the building also has different lights.
In my opinion, it's my favorite ride at DLR, so you should try to go on both versions. Wait times are high for daytime but not totally insane-basically if it's confined to the garden outside and doesn't extend to a queue in the roadway, I would wait. For after dark there are insane lines and I probably wouldn't do it without a FP. As PP mentioned, MaxPass is the best option to get a FP!