Keep in mind that there are the species Post Oak and White Oak. Both are members of the white oak group. All species of oaks fall into the two groups - red and white oak. Within the white oak group are numerous species including White Oak, Post Oak, Burr Oak and a couple dozen other species. Within the red oak group, we find Red Oak, Black Oak, Live Oak, Pin Oak, and a couple dozen other species. All white oak group species have similar wood, while all red oak species have similar wood. The biggest difference between the woods of the red and white oak groups are that when the wood dies, the end grain pores of white oak become clogged with the formation of toeless, while red oak remains ports (open pores). Take a six-inch long plank of white oak and try to blow through the end grain - you can't. Do the same with red oak and you can blow slowly through the plank of red oak. Better yet, put some soap and water on the opposite end and blow - you can blow bubbles out the far end!
Keep in mind that when you buy a board of "red oak" from the lumber yard, it may well be the specie Red Oak. But just as likely, it will actually be any one of the members of the red oak group - Black, Burr, Pin, etc. oak. Same with white oak lumber.
So, to answer your question directly: Post Oak is pretty much the same as White Oak (the specie) because BOTH are members of the white oak group! From the standpoint of wood for smoking, I have little doubt that Post Oak and White Oak perform much the same. It might perform a little bit differently than members of the red oak group, but even that I doubt it. I burn red oak wood (mostly Live Oak - a member of the red oak group) in my smoker because that is what is commonly available here in the Tampa Bay, Florida area.
My educated guess (B.S. in Forestry) is that for smoking meat purposes, any member specie of the white and red oak groups will be indistinguishable from one another. I wouldn't reach across to the other end of the wood pile for one type of oak over another.
Just fire that puppy up with what you have and enjoy!