<p>How is profiling different from logging?</p>

<p>Is it just that profiling is used for performance measurements to see how long each function takes?  Or am I off?</p>

<p>Typically, how are profiling libraries used?</p>

<p>What types of stats are obtained by profiling?</p>

I see profiling as performance measurement, and you don't have to profile every piece of code running, it's sometimes better to target specific areas.
Logging is storing information for later use, information which may relate to profiling but not necessarily.  It may just be to log what happened.
All the profiling stuff I've ever used basically allows you to store a start time and end time for a 'transaction' and later manipulate the data to see what's taking the most time.