After noon is quite vague. Assuming the context of the 24-hour day, it could mean any time in the day after 12:00 pm, whether that means 12:30, 3:00, or 7:00.
EDIT: As Edwin Ashworth pointed out, informally, after noon probably means a short while after 12:00 pm, maybe within the hour. However, if Joe says "I'll pick you up after noon" without specifying the upper bound on the time, it's still a very vague phrase. I would avoid concluding anything from such an interaction without further clarification, unless you want to butt heads about semantics when Joe shows up at 3:00 and says "hey, I wasn't lying!"
Afternoon is a noun that is defined as a specific time period. (TFD: The part of day from noon until dinnertime or sunset.) After afternoon, there is evening, and after that, there is night, both of which occur after 12:00 pm, but are separately defined time periods.
In essence, afternoon is after noon, but just because it is after noon does not mean it is afternoon.