Very approximately a heavier frame will be stiffer than a lighter frame, regardless of whether it is made of aluminium, titanium, steel.
If the Young's Modulus of a material is taken as the stiffness of a given material then it is possible to factor in the density of the given material to get an idea of the stiffness to weight ratio.
http://reynoldstechnology.biz/assets/pdf/rtl_steel_alloys_extract.pdf
Titanium is stiffer than aluminium, but titanium is more dense.
There is the possibility that the idea of springy titanium comes from the difficulty of cold working titanium, where there can be spring-back.
http://www.timet.com/fab-p24.htm