TCAS is sort of a fallback, last-resort system used in case all else fails. TCAS works within the scope of what happens within the next minute or so, and will require the involved aircraft to perform fairly extreme maneuvers to avoid a collision. TCAS has one single purpose: to ensure that two (or more) aircraft that have already gotten much too close will not collide. TCAS responds to a single, imminent danger that should not normally exist.
In contrast, ATC is a hugely complicated system that keeps planes moving safely and efficiently through the complex network that is our air infrastructure. Air traffic is carefully planned by ATC up to several hours in advance, and clearances and instructions issued by ATC ensure that planes are kept well clear of each other, so that close encounters do not happen at all.
As an analogy to road traffic, you can think of ATC as a set of traffic lights controlling an intersection, by taking input from a number of sensors and showing red/amber/green lights. TCAS on the other hand, is more comparable to the emergency auto-brake systems that many modern cars have, that will stop the car if, for example, a pedestrian suddenly crosses the road in front of the car.
So why does TCAS take precedence? Because in order for the TCAS to have been triggered at all, something else has already gone wrong - the main system has somehow failed, so the fallback system steps into effect. Looking at our road analogy, suppose you have driven into in intersection because you have a green light. If a car suddenly appears in front of you, you want the emergency brake to kick in, even if the light is indeed green and you should be allowed to go.