The principal argument against assigning authorship to Apollodoros comes from Carl Robert, De Apollodori bibliotheca (1873). There are a few points, and your own taste may determine which of them you find most compelling.
Apollodoros flourished in the mid-100s BCE, but Bibliotheke 2.1.3 cites Kastor's Chronicles, a work which covered events up to at least 61 BCE.
Apollodoros is best known for a work called Concerning the gods, which is the most likely reason that the Bibliotheke suffered upward attribution: the general topic is similar. However, their approaches couldn't be more different. The Bibliotheke is a straightforward summary of myths, taken at face value; what we know of Concerning the gods indicates that it was heavily rationalist, explaining beliefs in the divine in terms of either euhemerism (treating mythical figures as altered versions of historical ones) or allegory (reinterpreting gods as personifications of natural forces, with heavy emphasis on wordplay; e.g. Hēra ~ aēr 'mist').
Upward attribution is an extremely common phenomenon whereby well known works of lesser known people are reassigned to more famous people.
These points have generally been more than enough to take the manuscript attributions to Apollodoros as spurious. As I said, Robert is the principal source for this, but a decent recent edition of the Bibliotheke should give the same arguments (perhaps with some further ones).