Titanium is about 2/3rd the density of steel and 2/3 the bulk stiffness of steel. Similarly, aluminum is 1/3 the density of steel and 1/3 the bulk stiffness of steel. The way to make up for the loss of bulk stiffness is to use sections of larger diameters/sections to handle bending loads and thicker sections to handle axial loads. In bicycle frames there are reasonable amounts the frame members can be enlarged; thicker sections aren't required as the axial compression of the members is minor compared with bending/twisting.
The bending stiffness goes up with the cube of the section while the weight goes up linearly for same thickness, so the stiffness to weight ratio ends up increasing by the square of the section.
I recall that early aluminum designs had large amounts of flex because makers tried to keep the same tube sizes; then came the fat-tube frames and that seemed to have made the market happy enough. I presume the same thing happened with titanium.
Aside from being much more expensive there isn't a lot of advantage to titanium in a bike frame. Where titanium really shines is that, though typically weaker than high strength steels, it retains that strength at higher temperatures that are above the softening temperature of steel. I also think it is non-magnetic, for what that is worth. It is also corrosion resistant, but many parts of the bike are not, so maybe if the goal is to have a 100 year old frame that is left out of doors the entire time?
Carbon fiber seems to have pushed other materials off the high-price cliff. They can be formed with continuously varying section, have very high stiffness to weight ratio, don't seem to fatigue, certainly not the same way that aluminum can.
Titanium frames are an interesting ox-bow in the development of bicycles.