Bayesian: Sampling from Truncated Distributions When would the rejection sampling method be preferred to the inverse CDF method for sampling truncated random variables?
And when would the inverse CDF method be preferred to the rejection sampling method for sampling truncated random variables?
 A: The latter half of this answer gives a justification for why the inverse CDF method would be undesirable to use: the CDF is expensive or impossible to evaluate. The downside to the rejection sampling method is that you can potentially draw a lot of unwanted samples if the area you are truncating to is sufficiently small relative to the whole support. See here.
Thus, to answer your questions..

When would the rejection sampling method be preferred to the inverse CDF method for sampling truncated random variables?

When the CDF is either (A) impossible or (B) more expensive to evaluate than just doing rejeciton sampling

And when would the inverse CDF method be preferred to the rejection sampling method for sampling truncated random variables?

When rejection sampling would require you to waste too much computational effort in generating unwanted samples.
How do you know precisely when one is more efficient? That will depend on the specifics of the distributions, algorithms used to sample them, the specifics of your hardware and programming setup, etc.
There are other sampling methods which get around some of these problems. See the same wikipedia article.
