Do Rabin Fingerprints have any advantages over CRC?
This is an expanded version of my comment above.  At the top of page 135 of the paper cited by the OP, the construction of the cryptographic CRC is defined  in the following words:


  $\ldots$ for any message $M(x)$ of binary length $m$ bits, $h_p(M)$ 
  is defined as (the coefficients of) $M(x)\cdot x^n \bmod p(x)$.


Thus, the definition of cryptographic CRC in the cited paper is
the same as what the OP claims the Rabin fingerprint
is. On the other hand, the cited paper says Rabin's construction
is essentially the same 


  $\ldots$ except for the multiplication by $x^n$ in the modular
  process


In other words, the OP seems to have interchanged the
definitions of CRC and Rabin fingerprint in his question.

The cited paper 
also seems to be implying (the phrasing is somewhat ambiguous)
that Rabin's fingerprint is weaker than the cryptographic CRC
defined above, which, by the way, is the same as the way CRC
error detection is used in communications systems, as I stated
in my comment.