Is Young's modulus a measure of stiffness or elasticity? Young's modulus seems like a modulus of stiffness. It tells us how difficult is it (how much stress is required) to produce longitudinal strain in a solid. It does not tell anything about how an object will react when the deforming force is removed.
How can one on the basis of young's modulus decide that steel is more elastic than a rubber band?
 A: In physics elasticity is defined as the ability of a material to resist a distorting influence and return to its original size and shape when the distorting influence is removed (source Wikipedia).
Young’s Modulus is the ratio of applied stress to resulting strain in the linear elastic region of behavior. Therefore, they greater Young’s modulus the stiffer  a material is, that is, the greater the materials ability to resist a distorting influence (applied stress). For this reason, given the above general definition of elasticity, Young’s modulus is also called the modulus of elasticity.
Due to the highly overall non linear behavior of rubber I believe Young’s modulus for rubber is usually quoted forces small loads. The values for rubber are much lower than steel meaning that rubber is much less able to resist a distorting influence than steel as would obviously be expected.
Given the above definitions of elasticity and Young’s modulus we would conclude that the “elasticity” of rubber is less than steel. It is admittedly counter intuitive because on the one hand we think of rubber bands as being highly elastic in the sense that they’re easy to stretch. But on the other hand if they’re easier to stretch that means they’re less able to resist distortion, which is consistent with the physics definition of elasticity and Young’s modulus. 
Hope this helps.
A: It is not true that steel is more elastic than rubber. Not in common language.
Yes, steel has a larger modulus of elasticity, Young's modulus, the ratio of stress to strain $Y=\varepsilon/\sigma$. This is in the region of elastic response as long as the deformation $\sigma=\Delta \ell/\ell$ increases linearly with stress $\varepsilon =F/A$. The response is assumed to be immediate (fast, but slower than the speed of sound). For metals, this is on the order of 100 GPa, similar values for iron and for steel - it takes a lot of force to make a small elastic compression or elongation.
Steel also has a larger elastic limit of stress, a steel rod can support very large tensile loads without permanent changes in its length.
This does not say anything about the range of deformations that are reversible. For steel etc the elastic strain limit is usually on the order of $10^{-3}$. For rubber the range of elastic stretching is much larger. That is why rubber is elastic in ordinary language.
Compressibility is a word that has similar meanings in physics and in the common language. In physics, it is the reciprocal of the bulk modulus which is closely related to Young's modulus. In common language, compressibility is similar to elasticity.
