They're pretty much the same.
If you'll look them up in Duden, you'll see that they are defined as each other:
http://www.duden.de/rechtschreibung/Augenblick = "Zeitraum von sehr kurzer Dauer, Moment"
http://www.duden.de/rechtschreibung/Moment_Zeitpunkt_Zeitspanne = "1. Zeitraum von sehr kurzer Dauer; Augenblick; 2. Zeitpunkt"
OK, so Moment means not only "short duration" but also "point in time", as in im Moment nicht "not at the moment" or diesen Moment festhalten "capture this moment".
I'd say it has two, closely related meanings, and only one of them matches "Zeitpunkt".
It can mean "the moment when ..." = the point in time = der Zeitpunkt.
But it can also mean in einem Augenblick = "in a moment" = in a trice, in a jiffy, in a very short period of time, to emphasise that something happened very quickly or lasted very briefly -- "Zeitpunkt" can't be used for this meaning.
For example, if someone says "einen Augenblick, bitte", they are asking you to wait "one moment, please" -- you can't use "Zeitpunkt" for this.
But in "Der Augenblick, als ich begriff, wie alles begonnen hatte" = "The moment when I understood how everything had started", you could also say "Der Zeitpunkt, an dem ich begriff, wie alles begonnen hatte" with a very similar meaning. It's a bit less explicitly "quick moment" than "Zeitpunkt" but similar.