There is one theoretical difference between Shamir's scheme and Asmuth and Bloom's scheme.
Shamir can be done in an informationally secure manner; specifically, if the nonconstant polynomial coefficients were chosen in a random manner (that is, from a uniform probability distribution that's uncorrelated to anything else the attacker can see), then someone with a number of shares that's less than the threshold gains no information about the shared secret. This is because, for any potential value of the shared secret, there is the same number of possible secret coefficients that is consistent with that secret value - if we assume that those coefficients were chosen uniformly, the attacker has no criteria to say that one set of coefficients is any more likely that the other.
In contrast, Asmuth and Bloom's scheme does leak some probabilistitic information. Now, any particular value of the shared secret is possible; however different values of the shared secret have different numbers of possible secret value $\alpha$. Assuming that the attacker knows the probability distribution that $\alpha$ was drawn from, he can then assign different probabilities to the values of the shared secret, should he choose to.
As Asmuth and Bloom's scheme does leak some probabilistic information, while Shamir's does not, well, that seems like a good reason to prefer Shamir's (especially since, as Louis says, Shamir's is actually easier from a practical standpoint)