Could an object be stiff but not rigid or vice versa?
When is each one used?
And what is the opposite of each of them?
In engineering mechanics a rigid body will undergo no deformation whatsoever under any amount of force.
In contrast, stiffness is a measure of how much force it takes to deform a body by a given amount.
In general terms something is said to be stiff when it has a high stiffness coefficient, i.e., it takes a lot of force to deform it; but notice that it does deform.
As the stiffness goes to infinity the body becomes rigid, i.e., no amount of force will be able to deform it.
In practice, nothing is truly rigid, but many times assuming a rigid body simplifies the math needed to analyze a system and when the stiffness of the body is high enough the effects of the assumption are negligible.
Therefore in practice, I reserve the use of the word rigid for cases where there is no possibility for bending (whether figuratively, or literally) and use the word stiff when I believe bending is possible, albeit only with a large amount of force.
As for opposites:
opposite of rigid – deformable : able to be deformed
opposite of stiff – flaccid : not firm or stiff