There seem to be a lot of mathematics based answers here that I'm finding hard to understand.
I understood the original question was concerned with the difference between best rate & best angle of climb.
My own idea to explain the differences would be that there are two different angles of climb. one is steep but slower, the other is shallow but faster.
Over a given time the steep climb will be higher but not so far "down range" the shallow climb will be not so high but further "down range".
Placing two 30deg right angle triangles on a table, one with the long "square" side down, the other with the short "square" side down.
The hypoteneuse is the time (identical), the horizontal is the airspeed, the vertical is the climbed altitude.
Experts please advise if this is bullsh*t.