A denitrifier basically removes nitrates in an aquarium system by using anaerobic bacteria to remove the oxygen in the denitrifier, forcing them to eventually removing nitrates from your system. In a coil system you have to cycle your coil in a similar way you'd cycle your tank so that the anaerobic/ anoxic bacteria have time to populate your denitrifier. The flow on these has to be just right and in my opinion a nightmare to adjust etc. You don't see these around too much anymore except for large aquariums etc.
A skimmer on the other hand, removes organic particles from your system before it has a chance to it decompose into ammonia, nitrite, nitrate etc. It's basically easing up the load on the natural filters in your aquarium (LR
, Beneficial bacteria) so that it can handle the rest. Nitrates are usually removed by plants and\or PWCs. Once dialed in, a skimmer needs no maintenance besides emptying the skimmer cup from time to time as needed.