Internal filters comes in many sizes, with smaller ones having smaller flow rates and vice versa. The smallest internal filters are mainly for mechanical filteration and the media compartment only contains a small sponge. Bigger internal filters sometimes come with more than one compartment for bio media and even chemical filteratiion.
One of the biggest advantage of internal filters is peace of mind. You know that no matter what happens, your aquarium will not get drained. (There has been cases in overhead and caninster filters where water can get channeled out due to pipe leakage or loosening causing a flood in the home and a drained tank)
Another advantage is that they usually come with an air line that can suck in air from above to infuse into the water so you do not need to run a separate air pump.
The disadvantage of internal filters is difficulty of maintainance. Most internal filters come with tight fitting filter compartments that are difficult to remove when used in the tank. This is not a flaw but is necessary because you don't want it to drop out by itself during operation. All internal filters vibrate to different extents, and during operation the constant vibrations, although minor, can detach a loose chamber unless tightly secured. Due to this difficulty, most people maintain their internal filters less often than they have to really should, leading to a buildup of scum in the filter media. When the filter compartment is opened, some of the scum escapes into the water column causing cloudy water. (Fortunately they are harmless and clears quickly). Some internal filter allows the whole unit to slide out of a frame with suction pads so that the filter compartment can be opened outside of the tank. This have its problem too, depending on whether you switch off the power supply.
If the power is switched of, filter scum can leak out from below due to the absense of suction force. Also after maintainance, when the filter is turned back on, loosened filter scum from the sides of the propeller and tunneling is flushed out into the water column. If the power is not switched off, firstly it is not a safe practice to handle it. Secondly, there is a brief period of time where the unit is running outside of water. In the long term, this is detrimental to the propeller.
Lastly, internal filters take up some of the swimming space in the aquarium and can be obtrusive aesthetically.