Revealing mistake: In the part in the motel where THING is doing morse code on the table to Gomez you can see the cut-out in the table where the guy's hand is sticking through the table.
Other mistake: When Gomez begins to flip through his desk calendar, we see that "today" is Thursday, March 28, 1991. He then turns to the date of the "next quarter". The left hand page shows Thursday, August 8, but the right hand page shows Tuesday, September 9. (00:11:15)
Revealing mistake: During the Mamoshka dance, right when Gomez throws the knife high in the air (the one that lands in Fester's mouth), the camera goes to a wide shot when the knife is in the air, and you can plainly see that the jugglers are Fester and Gomez's stunts.
Answer: Presumably when Fester was lost in the Bermuda Triangle, whatever supernatural forces caused him to lose his memory also made him grow hair. This gets reversed at the end of the movie when he's zapped by lightning, restoring his memories and destroying his hair follicles as well.