This is a great question that I pondered myself while getting my multi-engine rating.
V1 is determined by several factors but the one most important factor is called the "accelerate / decelerate distance". In other words, the distance it will take to stop the aircraft before the aircraft runs out of runway or continue safely into the air.
Ordinarily you might think V1 is the same, but "lower" performance and "higher" performance aircraft behave differently during takeoff. A comparatively low performance aircraft such as single engine recip will have V1 and VR equal. But high performance aircraft such as a turbo prop or jet will have a V1 quite different from VR.
Acceleration is a speed - or more accurately a change in speed over time designated by delta-a meaning that an aircraft accelerating before stall speed will continue to INCREASE in speed for a short period of time even if there is an engine thrust failure - in layman's terms this is called momentum.
Therefore, in moderately or high performance powered aircraft V1 and VR will be different and the manufacture will base the value on flight tests.
Lets consider two examples. A single engine airplane like like a Cessna 150 will have V1 and VR at the stall speed. so the pilot pulls back at V1 and by the time the aircraft lifts off the ground it has gained an additional 5kts an climbs at V1+5kt. In a jet, the pilot will reach VR 10 or 20kts before V1 but even if the engine fails, momentum will carry the aircraft past V1 and safely the aircraft will reach flying speed without additional thrust.
Therefore, depending on what we might call excess (yes a slightly inaccurate term) thrust, VR occurs prior to V1 because of delta-a (high acceleration).
For more powerful aircraft, VR is a moving target. B747 have so much power that if they are light weight or have a long runway, they are authorized to use less than full power for the takeoff to conserve engine wear and noise abatement. Therefore VR is different for each takeoff.
Update: As noted below, for the single engine scenario I should have used VS not V1. And, yes powerful aircraft have a Delta-A - change in acceleration, that is different from their velocity. As an example, earths gravity (32ft/sec/sec) is an acceleration not a pure velocity.