The major difference between the two engines (1ZZ in the XR and base Matrix, vs 2ZZ in the XRS) is in how the valve train behaves at high RPM.
The 1ZZ engine has a fairly normal valve train. The VVTi system varies when the intake valves open and close to give high performance when you need it, and good fuel efficiency and clean burning at other times. The cams push the valves open in the normal way as the VVTi system just varies the position of the intake cam. The exhaust cam is not variable, which is also typical.
The 2ZZ engine, on the other, has a trick up its sleeve. At high revs (above 6000 RPM) it slips a shim into a different position for each valve (powered by oil pressure, controlled by the engine computer) that causes the intake and exhaust valves to open earlier, open further, and stay open longer. This allows the engine to handle a LOT more air and fuel. They call it "lift", because it lifts the valves up higher (or lower, depending on whether or not you are standing on your head), and I'm told that when you hit "lift" you really get thrown back into your seat. The torque curves for the 2ZZ engine show torque really jump in the neighborhood of 7000 RPM, whereas the 1ZZ engine torque has really fallen off even by 6000 RPM.
Below 6000 RPM the two engines behave similarly. The 1ZZ actually has slightly better torque in that range, but the XRS (with the 2ZZ) is geared to rev higher so actually has more torque to the wheels (with slightly higher engine revs).
Also, because the 2ZZ engine has slightly higher compression, it requires premium fuel, whereas the 1ZZ takes regular.
If you don't plan to really make the car scream, the 1ZZ engine is the better option for most people. If you want the thrill of really screaming in a car that doesn't cost an arm and a leg, the 2ZZ is the way to go.
I drive an XR: 2nd gear to 60 MPH (6000 RPM) gets me onto a highway as fast as I need to go!