As has been commented, Asmuth-Bloom is does not always give a perfect scheme. The original paper gives a condition on the primes to maximise "sharpness", which is what they call their measure of closeness to perfection.
A recent paper at Asiacrypt 2018 gives fairly detailed analysis of the differences, and they also construct a new scheme using similar techniques. In the introduction they remark:
In a word, Shamir’s scheme is ideal and easy to construct
while Asmuth- Bloom’s scheme is not ideal, hard to construct
but more natural and neat in constructing weighted SS scheme.
So while Shamir's scheme is very good for threshold schemes by most metrics, if one wants to construct secret sharing schemes for weighted threshold access structures then Asmuth-Bloom, these authors claim, has a more "natural" generalisation. (Weighted threshold schemes are defined in the same paper, and are basically a way of fine-tuning the qualified sets of the access structure.)