Both seem to be appropriate for timing things on Android. How are they different? Which one is best for timing?
As far as I can tell, there's not really much of a difference, but they use different Linux clocks under the hood.
nanoTime() uses either clock_gettime() or gettimeofday(): see the code here (part of Dalvik VM).
elapsedRealtimeNanos() used to use clock_gettime(), but it no longer does. Instead, it now issues an ioctl to the /dev/alarm device (see code here), but I'm not exactly sure what the alarm device driver uses.
I've used elapsedRealtimeNanos() for a long time with no trouble, but not for any specific reason. Just make sure you realize that neither of these functions will provide the wall clock time.