Sponge filters:
Pros
- Gentle, generally only creating an upward current
- Do create surface movement (good for getting dissolved oxygen into water, though that isn't important for bettas)
- Can be adjustable with a check valve in the airline
- Come in a big range, from tiny and wee to huge and monstrous
- Extremely easy to clean (just rinse sponge in dirty tank water)
Cons
- They don't usually come with the airpump needed to run them, so you may need to buy one
- They create a bubbling noise that you may find annoying
- There is frequently only one sponge in smaller ones, so replacing it will restart the cycle
Internal filters
Pros
- Huge variety of brands and styles
- Usually easy to baffle
- Easy to maintain
- Multiple stage filtration
- Some are very quiet (no waterfalll noises either)
- If one part needs replacing, you won't lose the cycle
Cons
- They take up room inside the tank
- Most brands mean you have to get your hands wet to maintain them
- During maintenance, you may get some gunk in the tank
External filters
Pros
- Come in two types - HOB and canister - for different purposes
- Some are easy to baffle
- Easy to maintain
- Multiple stage filtration
- Some are very quiet
- If one part needs replacing, you won't lose the cycle
Cons
- Some are not so easy to baffle
- HOB filters require space on the back of your tank
- Can ruin the asthetic of the outside of the tank
- Some make annoying waterfall noises
That's my summary.
I haven't touched on Undergravel filters because I hate them.