The headband of intellect (Dungeon Master's Guide 258) (4,000+ gp; 0 lbs.) really does not grant extra skill points: The item's description says, "This [magic item's] enhancement bonus does not earn the wearer extra skill points when a new level is attained; use the unenhanced Intelligence bonus to determine skill points." However, the enhancement bonus a headband grants to a wearer's Intelligence score affects just about everything else involving the wearer's Intelligence score.
The Greyhawk regional feat Faerie Mysteries Initiate (Dragon #319 58-9), for example, grants the feat's possessor, when the passions are experienced, the ability "to use [its] Intelligence modifier instead of [its] Constitution modifier to determine bonus hit points," so a headband would affect these bonus hp, but any extra hp would be lost were the headband removed… and more hp would be gained were the wearer to replace his old headband with a headband having a greater enhancement bonus.
Skill modifiers for skills with Intelligence as their key abilities are affected just as if the creature's Intelligence score were actually higher. Likewise, the enhancement bonus can affect a wizard's saving throw DCs for his spells and grant the wizard more bonus spells due to a high Intelligence score.
Another thing besides skill points that is not affected by a headband's enhancement bonus is bonus languages due to an Intelligence score of 12 or higher: "A smart character (one who had an Intelligence bonus at 1st level) speaks other languages as well, one extra language per point of Intelligence bonus as a starting character" (Player's Handbook 12). Most characters don't start at level 1 with a headband, so a headband earns the typical wearer no bonus languages.
Note: Prior to the 3.5 revision, the line in the headband description about not getting extra skill points is absent. I thought I remembered a Sage Advice suggestion that a creature gain the skill points upon advancing a level due to an increased Intelligence score from the headband if the headband were worn continuously for the majority of the previous level, but I can't locate this suggestion now. Further, Pathfinder grants skill ranks to the wearer—in a totally different fashion—of its renamed headband of vast intelligence.