(It is assumed that Pearl was a servant on Homeworld, so look for other theories explaining that.)

Before "Catch and Release", Peridot's gem tech hands were used for data and weaponry. It was the only thing Peridot relied on before having them taken away. When her bubble got popped, she tricked Steven so she could run. If she realized that her robotic hands are gone, she'd would've gone to summoning her weapon much like the other gems. Maybe Peridot wasn't allowed/couldn't be able to use her gem on Homeworld since she was a servant.

If we could figuratively label the mentioned classes on Homeworld, then from lowest to highest it will be...
-Servant
-Informant
-Escort
-Diamond

Based on assumption, the Servant Class on Homeworld weren't allowed to use their gem which is why Peridot never used her gem. When Pearl (also a servant) came to Earth, she learned to summon her weapon with the flow of petals from her tree.

Pearl Servants confirmed in "Back to the Barn"!