Chris - slew rate is a slope of the signal going up or down expressed in V/us (rate of change). Rise time is only for known amount of volts going up (fall time for going down).
We use slew rate when voltage doesn't have any particular range of interest (output of an amp etc.) and we just want to know how fast is it changing. Another case might be when something depends, by definition, only on the rate of change - like transmission line effect.
Rise time and fall time are used mostly when voltage range is already defined like 5V logic or 3.3V logic etc.
Slew rate (as well as rise or fall time) can be applied to any analog quantity like current, resistance, light etc.
