External pros
More space for media/sponges
Easier to maintain (generally)
Higher flow rates are available
No clutter in the tank
Able to choose what order your water is filtered in
External cons
Usually pretty big
Much more expensive
Media/sponges usually cost more
Internal Pros
Smaller, ideal for smaller tanks
Cheaper running costs
Some are pretty easy to maintain
Cheaper to buy brand new
Internal Cons
Lots of clutter in the tank
Sometimes, unable to change part of the media and can only do full changes, causing a new cycle (EG: the Interpet's PF range of internal filters)
Can be a bit fiddly when maintaining (shouldn't be an issue with more modern filters though)
There's probably LOAADS but I'm pretty tired
these are what I can think of off the top of my head though. In my experience, I'd ALWAYS use an external.