From the standpoint of most critics of these philosophies, there is no difference. An analogous situation is the diversity of thought that falls under the umbrella of Marxism:
communism, socialism, Leninism, Bolshevism, Trotskyism, Maoism, and even state monopoly capitalism. To a critic, all of those categories are meaningless since they reject at some level or another the principles of capitalistic economic theory. They are distinctions of no import to a critic.
Obviously, a proponent of one of these subphilosophies (hierarchically speaking) would make many distinctions between their positions and others which from outside seem similar. It's the nature of the beast.
As someone wholly outside, it seems to me that the difference is whether the goal is to alter humanity to some specific endpoint (posthumanism) or whether simple improvement is in view (transhumanism). It seems likely that the later used term is a correction, expansion, or clarification of the earlier term. (This seems to be the pattern: Platonists are followed by neo-Platonists. Marxism is followed by Marxist–Leninism.)
According to Wikipedia, "transhumanism" was coined in 1957. The original definition of transhuman according to Julian Huxley:
The human species can, if it wishes, transcend itself —not just sporadically, an individual here in one way, an individual there in another way, but in its entirety, as humanity. We need a name for this new belief. Perhaps transhumanism will serve: man remaining man, but transcending himself, by realizing new possibilities of and for his human nature.
"Post-Human" was coined no later than 1988. Here's Robert Pepperell's definition:
2.12. First we had God, Humans and Nature. The Rationalists dispensed with God leaving Humans in perpetual conflict with Nature. The Post_Humanists dispense with Humans leaving only Nature. The distinction between God, Nature and Humanity does not represent any eternal truth about the human condition. It merely reflects the prejudices of the societies which maintained the distinction.
So the general idea of humanity pulling itself to some transcended state seems to be a common thread, but whether the result is still human seems to be at issue. If I've got the order of coinage correct, all posthumanists owe their philosophical heritage to transhumanism, but not the other way around.