Well they are 3 different things. For your tank, mechanical filtration would be the pre filter sponge, chemical filtration would be the activated carbon in the filter and biological filtration is your live rock/sand.
Mechanical filtration generally removes larger particles from the water.
Chemical filtration somehow cleans the water through a chemical reaction in some sort of filter media i.e. activated carbon removing the yellow tinge of water or granular ferric oxide removing phosphates in the water.
Finally and probably most importantly is biological filtration. These are the nitrifying bacteria that make your tank go through nitrogen fixation (or the nitrogen cycle which is basically cycling your tank) and turn Ammonia excreted by fish and other inhabitants into nitrate. This is constantly happening. You remove excess nitrates with partial weekly. water changes.
The Advantages and disadvantages are all different, but for are needs usually when you first set up at tank you will want all 3, then as time goes on get rid of mechanical and then some more time the chemical can go, then you will have a fully established tank that can run on the normal bio filters the tank creates. Here's my short and sweet opinion on them.
Adv. for mechanic would be being able to take out large chunks of debris fast.
Dis. Adv. would be cleaning very regularly and if not cleaned enough will cause high nitrates.
Adv. for chemical as said before takes the yellow out of water if you have that and taking out phosphates, and many other things like some nitrates and all different thinks depending on the type of chemical filter.
Dis. Adv. would be having to buy more and change it too, and wrong type for saltwater can hurt things, and some some kinds of chemical filters can cause debris to build up causing high nitrates.
Adv. for bio is above all the best IMO being able to take out nitrates and break down all bacteria and having clean clear water, with no maintenance need really at all. just care to not kill the beneficial bacteria.
Dis. Adv. would be large chunks of debris will be present until broken down, and if something done in-proper happens the bacteria could die, such as power head falling in sand bed stirring it all up and making corals unhappy and taking out live rock to do unknown testing things like washing it off as my dads done before.
If you are interested in learning about reefkeeping or have a problem with your reef, our reefkeeping community is here to help. Feel free to ask a question or search our site. We have lots of experienced reefkeepers that are willing to provide free reefkeeping advice!
Besides being a great resource for all levels of reef aquarium hobbyists, CaptiveReefs is a social experience that will enhance your enjoyment of reefkeeping. CaptiveReefs is committed to connecting reefkeepers with the support and information they need to grow beautiful coral reef aquariums.