There's a difference betweeen 'to know' and 'to understand.' Yet we commonly say:
"I don't know what you mean."
A bit nitpicky, but I wouldn't be inclined to use "I don't understand what you mean." It's a bit too pleonastic for my taste. I would either say: "I don't understand." Or "I don't know what you mean."
"I don't get it." will often work, but not always. To 'get' something is like what the Germans call an 'Aha-erlebnis,' a moment in which suddenly everything is clear. So, when someone tells you a joke you don't understand, you'd say: "I don't get it!" Yet when someone addresses you on the street, in a language you don't know, "I don't get it." doesn't work at all. You'd say "I can't understand you." Or "I don't know what you're saying," etc.
People sometimes say things like "I don't get math." Which is not entirely the same as "I don't understand math." To say you don't 'get' math is a way of expressing a thought like: "The quarter on this whole math thingy isn't dropping for me." Meaning: you're looking for that Aha-erlebnis, for that moment of satori, at which time you will suddenly 'get' how it is all done, but it's just not coming.