The big advantage of Titanium over the other materials is it's strength, weight and durability. It's an almost ideal compromise. The downside is that it isn't as indestructible as you might have been led to believe, and it tends not to be overtly characterful. A high-end steel frame for example will have a very noticeable (and charming), liveliness to the ride, while a good carbon frame will give a bizarrely smooth and comfortable experience, yet still provide great agility and control. Titanium blends these in a noticeable but not overt fashion. In a sense it's a neutral material.
That makes it sound bland and dull, but in fact it can generate a tremendous amount of confidence on the bike. It seems to remain composed and exibit a degree of stability in extreme conditions that's hard to define exactly. It's extremely springy, but in a contained manner and that allows for a surprising amount of technical control. If it has a unique ride quality, then it's a certain supple 'meatiness' that isn't quite there in steel and carbon.
So it's better than steel in that its lighter and won't rust, and it's better than carbon in that it won't damage quite as easily (chainsuck for example), and it blends the best ride characteristics of both. Other than that, good steel and good carbon are as good as good titanium.