Don't they both mean lacking a specific color, like orange, yellow or red?
To state in the most basic of definitions...
Anerythrism is the lack of red pigmentation. Yellow pigmentation, if generally found in the species, can either remain intact or develop over a period of time. This is why when you see, for example, Anerythristic Corn snakes, they will usually have a certain amount of yellowing on their cheeks and necks. The same amount of yellowing can also be said for Snow Corn snakes which is basically the crossing of an Anerythristic to an Amelanistic animal.
Axanthism is generally about the lack of yellow pigmentation but it can also include the lack of red pigmentation as well in some species of animals. Although some Axanthic animals can display a brownish color, it has been suggested to me by Charles Pritzel, author of the Genetics For Herpers book, that the brown is possibly a separate color pigmentation and not the result of yellow pigmentation being mixed with the black pigmentation. This of course makes sense to me after studying and trying to determine all the different colors and pigmentations that can go into making a brown color. I would also believe that if yellow pigmentation did indeed exist, then there would be strong evidence of that found on the ventral scales as well where solid yellow coloration is often found, which, from examination of my animals, it is not.
I've seen these guys this week advertised on KS and I can't tell the difference between an Axantic or a regular hog in some of the pictures.
I will assume you are referring to the ad(s) placed by Jeff Nemanius/Gateway City Reptiles about his Axanthic for sale. If so, I do agree that pictures of one particular animal do not look Axanthic to me either. This of course isn't to say that the animal isn't Axanthic, but it definitely lacks the typical coloration of similiar animals in it's age group. I would hesitate to speculate any further and leave it up to Jeff to come here and explain. 
Is this one of the situations where it takes a year or so for the full color to come out?
Again, this all depends on the individual animal. I bought my female Axanthic from BHB which was black and silver at only several months old. She eventually lightened up to almost ghost like colors before darkening up again. She has never exhibited any brown or yellow pigmentation. On the other hand, our male Axanthic that we purchased from Jeff Nemanius actually had quite a bit of brown in him, especially on his head. After several months, he eventually shed off the brown pigmentation and is now a very dark black and silver Axanthic. Quite different in contrast to our BHB female.
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