The abilities of thieves can be useful, the skills of a thief in sneaking an security allow you to get at things others might have trouble getting to, as you may be able to slip items away without alerting anyone nearby, or unlock locked doors and containers. Their abilities are also useful in dealing with the enemies that populate many parts of the game, a thief could try to sneak past enemies instead of attacking them, or if you must kill an enemy, you could use stealth to gain a sneak attack bonus to damage. Though I find being a stealthu character more fun in Oblivion than Morrowind, aside from the stealth system being a little better, the Thieves Guild quests were also more interesting, Radiant AI also added to being a thief, as you could wait around until the owner of a house or shop was asleep or gone, then break in and steal things, much more intereting than walking up to the completely unguarded second floor and take things without any effort, or the owner even coming to see why some shady looking Khajiit walked in his front door, went upstairs, and spent an unsual amount of time up there. On the down side, without mods, there isn't all that much that's really worth stealing in Oblivion, and you can only sell anything you steal to specific merchants, in Morrowind, you could sell stolen goods to anyone except the person you stole them from, and some places had some pretty valuable things to steal, especially the great house vaults (which, sadly, if you were strong enough to kill a few ordinators and persuasive enough to taunt them into attacking you, you could rob without being a skilled thief and not get any bounty.)
As to whether I use thief skills, it depends on the character I want to play at the time, though even if I make stealthy characters, still take a few weapon skills, as even if you sneak past every enemy you can, at some point, you'll likely have to kill someone or something.
Both must be stealthy, remaining unseen. At least that's how I look at it...
That would be it, assassins try to kill people and thieves try to take things from people, both often rely on stealth to do this.
Umm.... I snuck around all the time in Morrowind. With a Redguard assassin, no less, and I hardly had unholy cheat-like sneak levels.
It is possible to sneak in Morrowind if you get your skills high enough, the only problem is getting your skills high enough, without training, I've found that raising the sneak skill can be quite difficult at lower levels, of course, it's not like you won't have enough gold for training.