Sponge filters remove harmful Ammonia in the tank, and convert them to nitrite (and then nitrate), the later is not/less harmful to the fish. It does absorb some little particles suspended in the water, as part of the filtering generated by the rising bubble air sucking in the water, but nothing compared to a power filter.
I do weekly water change by siphoning out 30-50% of the water. This will remove the waste that accumulate at the bottom of the tank and also remove the final waste product in nitrate form. Every few weeks, I follow the water changing with removing the sponge filter itself, and squeeze it inside the bucket of water I just siphoned out from the tank, in order to get rid of particles trapped inside the sponge filter.
If you use a power filter, it basically does the same with the additional benefits of also trapping suspended particles. The bio filtering is done by the bio-filters inside the power filter, and the power filter also trap suspended particles in its mesh filter. With that said, you still have to do weekly/bi-weekly water change to remove the nitrate.
The problem with using power filters is that the betta do not like moving water, and may harm their long fins, so that's why sponge filters are preferred.
For more infos about the nitrogen cycle, here: