Obviously a tail number is used to distinguish yourself from other aircraft, but after the first call, a callsign does the same thing. It is also used for investigation purposes I would imagine, by listening to radio recordings, but are controllers typically even able to read tail numbers from the tower or do they just see a plane around where you say you are calling from and assume that it's you?
Sometimes your callsign is your tail number, particularly on General Aviation flights. For example, if you are flying the Cessna 152 with tail number "N123AB", you would call ATC with "November-One-Two-Three-Alpha-Bravo".
Commercial flights, on the other hand, tend to use their flight number as the callsign. E.g. if an aircraft is operating for Delta Air Lines flight 123, the callsign would be "Delta-One-Two-Three". This is not applicable to GA as GA do not usually fly a particular route or for a particular company. Assigning a flight number to every GA flight would be troublesome, it is simpler to just use your tail number as it is guaranteed to be unique.
Controllers may identify a plane by tail number when you are on the ground, but once in the air it is almost impossible to sight a plane this way. If this is your initial call to the controller, you would describe your location, which allows the controller to identify you on the radar screen. If the controller is unsure, you may be assigned a squawk code and even asked to push the IDENT button. You may also describe your type to the controller, like:
"Tower, N123AB is Cessna 152, 5 miles South of XYZ, (...)"
You are correct that if you do lie about your callsign, ATC as no way of knowing it unless somebody sees you, as described in the "Inside Air Force One" documentary by National Geographic when they tried to pretend to be a GulfStream aircraft to secretly transport the president. If you have ADS-B installed ATC will be able to see your callsign as it was programmed into the ADS-B unit, but even that can be re-programmed.
Check out the image on the bottom of this page: http://www.oocities.org/rjt02/behind.htm . For a general aviation aircraft (as the above poster described), when you contact a controller from the air, you'll tell him what you are (Cessna Skyhawk), who you are (N123AB), where you are (5 miles south east of XYZ airport at 6,000 feet), and what you want (inbound to land at XYZ (or) requesting flight following to JKL airport (or) transitional your airspace (or) requesting Bravo clearance etc).
The controller will look on his radar, which shows not only the position of aircraft, but also the altitude and squawk code. When you tell him the tail number and type, he can edit that blip to add that information.
Callsigns are useful for a few situations. First of all, the controller will know instantly that Delta 1234 is heavy and fast. Similarly, they can identify some special operations by name (SAR missions, military aircraft, police helicopters, skydive ops).
It's not plausible to give every aircraft its own callsign, but for aircraft that a controller interacts with frequently, and operate differently from the norm, it's another tool allow ATC to manage mental workload.
Air Traffic Control primarily provides aircraft separation services to aircraft operating under Instrument Flight Rules (IFR). In US airspace, aircraft operating in US airspace under Visual Flight Rules (VFR), flying in and out of un-towered airports are not necessarily required to identify themselves to ATC. VFR traffic does not receive ATC separation services, they are expected to see and avoid other aircraft and obstacles.
ATC will know the registration number of all aircraft they provide IFR separation services to from the IFR flight plan. The radio call sign may be the aircraft registration or the Flight ID (in the case of an airline).
In order to provide efficient and safe use of the airspace for IFR traffic, ATC does employ various aircraft surveillance systems. One of those systems relies on onboard Mode S Transponders. The S stands for selective or addressable.
Every aircraft registration number (World Wide) can be translated to a unique ICAO 24 bit address. If an aircraft is equipped with a Mode S Transponder, every transmission of that transponder will contain the 24 bit ICAO address that maps to the aircraft registration. General formats for the ICAO 24 bit address are defined in ICAO Annex 10 Volume 3, Chapter 9.