Do Rabin Fingerprints have any advantages over CRC? Background

In both, bitstrings are interpreted as a polnomical over GF(2) and they each can be used to implement a hash over a sliding window. The definitions of each are as follows:

Rabin Fingerprint: $M(x) = Q(x) \cdot G(x) + R (x)$

CRC: $M(x) \cdot x^{n} = Q(x) \cdot G(x) + R (x)$

Where $M(x)$ is message polynomial, $G(x)$ is the degree-$n$ irreducible 'generator' polynomial, $x^{n}$ represents $n$ zeroes added at the end of the message, $Q(x)$ is the quotient polynomial (ignored) and $R(x)$ is the remainder polynomial (the hash itself).

The Question

According to page 7 (labelled as page 135) of the paper LFSR-based Hashing and Authentication, CRC offers some improvements for hashing under encryption.  

Assuming $G(x)$ is the same for the implementation of each algorithm (i.e. ignoring that polynomials under Rabin should be random) and an identical $M(x)$ is used for testing, does the Rabin Fingerprint offer any advantages over CRC? If not, is there any reason why the Rabin Fingerprint is moderately popular in file chunking (using the sliding window technique mentioned above)? For example, 1, 2 and many others on the web. By contrast, I can find very few implementations of a sliding window version of CRC or mentions of it being used for chunking.
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.