What is the difference between Best Rate of Climb and Maximum Rate of Climb?
Generally when GA pilots talk about climb performance we speak of two different airspeed values:

Best Rate of Climb speed (Vy) gets you the most altitude per unit time (feet per minute).
When you want to get to cruise altitude quickly for maximum efficiency you'll aim for the best rate of climb so you spend the least time at lower, less efficient altitudes.  

Best Angle of Climb speed (Vx) gets you the greatest altitude per unit of ground distance (feet per mile).
When you've got a FAA-Standard 50-foot-tree at the departure end of the runway you'll aim for the best angle of climb to ensure you don't wind up in the tree.

Those speeds are useful to us as pilots, but the exact rate of climb (feet-per-minute) for those speeds will vary: A fully loaded plane will climb more slowly than one that's just got the pilot and a few gallons of fuel on board, and that's where the "maximum rate of climb" enters into the discussion:

Maximum rate of climb is the number of feet per minute you can get climbing at the "best rate of climb" airspeed.

If someone is being sloppy in their usage "Maximum Rate of Climb" could mean "Best Rate of Climb" (pitch for Vy and you get what you get), but if you're being precise in your usage and really talking about the rate of climb it would mean the theoretical maximum rate of climb in feet per minute based on the current conditions and aircraft weight.



Maximum rate of climb under a given set of conditions is useful information to know if you need to clear terrain at some point on your flight path and want to be sure you can climb fast enough to do so: If you're starting from sea level and need to clear a 5000 foot mountain that's 5 minutes away but your plane can't manage more than 500 feet per minute at best-rate-of-climb speed under the current conditions you'll need to reconsider your flight plan to either avoid the mountain or climb in a circle somewhere until you can clear it.