Specalists can only carry one primary. You can buy a slot license which allows you to use two primaries. (Slot A and Slot B)
Specialists come with a specialist slot which allows you to use a special weapon. (i.e., flamethrower) you can buy another special slot, similar to the primary slot license, which will allow you to carry two special weapons.
Specialists cannot equip gear. (I.e., Cyclops Bandana)
Specialists will usually get kicked for using their special weapons. (i.e., flamethrower in a Quarentine Game or in FireTeam)
Mercenaries are characters with added abilities. (I.e., speed increase, protection increase) Unlike specialists you can equip gear, (i.e., Cyclops Bandana) with the exceptions of vests. Some characters may allow you to use vests but I am unsure of which ones.
Mercenaries allow you to use any weapon that you could normally use. You cannot buy and use special weapons.
Mercenaries will, most likely, not get you kicked for being a noob compared to a specialist.