The Bear is a very important "monster" in the OSR, because it is a "teaching" monster. Sure you could fight the bear and perhaps lose a party member or two, but it's much better to avoid it or bribe it with food instead. Particularly devious parties will even trick the bear into fighting their enemies for them. Learning how to deal with a bear is an important rite of passage for gamers.
(I'll note that the bear stats can be re-used for a *lot* of monsters, but that's a topic that has already been well covered by other bloggers).
Ok, so now your players know how to deal with a bear. Can you still use bears as a GM or is it boring? Of course you can!
The local villagers approach the party because they are having a bear problem. What is going on?
1. A single bear has been spotted by villagers in the woods; it is acting erratically and looking "not natural". The bear is actually a bear-shaped robot piloted by tiny space gremlins who are exploring the local fauna and flora. They didn't do a good job disguising in their exploration craft.
2. The bear has been eating moths in the mountains to fatten up, as bears do. It has found a large batch of dream moths. He is now stalking the dream world and giving the locals nightmares.
3. An alarming number of bears have been spotted in the woods and people are disappearing. A virulent case of bearantropy is to blame - merely being in contact with the bear is enough to catch it, with an onset of mere minutes. The PCs must found the original carrier and cure it to break the curse so they can return everyone (and possibly themselves) to normal.
4. An alarming number of bears have been spotted in the village. The 14 bears are intelligent and armored. They are a mercernary band on break between jobs, and they are eating a *lot* of food. The locals are very nervous
5. A bugbear had a bad encounter with ogres and has lost all its belongings, including its clothes. It's quite embarrassed by the whole situation.
6. A wizard has been polymorphed into a bear and is trying to get help to turn back into her original form.
7. A particularly clever local bear feels it has been born into the wrong body and dreams of being human. It is trying to fool humans into turning it into an humanoid.
8. There is no bear. The locals are cannibal cultists who wanted to lure the PCs into the wood to hunt them down and eat them. The forests is filled with traps and ambush points.
edit/PS: someone said I should do tigers and bears. Well I can't possibly do tigers better than this. Brace yourself! http://goblinpunch.blogspot.com/2013/11/inventing-tiger.html