What are the advantages, if any, of derivative sampling?
Papoulis introduced a generalization of the sampling theorem [1], of which derivative sampling approach is one case. The gist of the theorem, quoting from [2] is:


  In 1977, Papoulis introduced a powerful extension of Shannon’s sampling theory, showing that a band-limited signal could be reconstructed exactly from the samples of the response of $m$ linear shift-invariant systems sampled at $1/m$ the reconstruction rate. 


Perhaps one reason why it's hard to search for the term is because Papoulis' generalized sampling theorem is mentioned more often than "derivative sampling". [2] is also a very good article which presents a broad overview of the sampling approaches at the time of publication. [3], also by the same author is an extension of [1] to the class of non-bandlimited functions. 

As for applications, in a recent paper [4], the derivative sampling approach is used to design wideband fractional delay filters and the authors show that sampling the derivative results in smaller errors. From the abstract:


  In this paper, the design of wideband fractional delay filter is investigated. First, the reconstruction formula of derivative sampling method is applied to design wideband fractional delay filter by using index substitution and window method. ... Finally, numerical examples are demonstrated to show that the proposed method has smaller design error than the conventional fractional delay filter without sampling the derivative of signal.


While there certainly are more, I'll refrain from posting more references and application to keep it short (and avoid it turning into a list). A good point to start looking would be to check which papers have cited [1]-[3] and narrow down the list based on the abstract.



[1]: A. Papoulis, “Generalized sampling expansion,” IEEE Trans. Circuits and Systems, vol. 24, no. 11, pp. 652-654, 1977.

[2]: M. Unser, "Sampling - 50 years after Shannon," Proceedings of the IEEE, vol. 88, num. 4, p. 569-587, 2000

[3]: M. Unser and J. Zerubia, "A generalized sampling theory without band-limiting constraints," IEEE Trans. Circuits and Systems II, vol. 45, num. 8, p. 959–969, 1998

[4]: C-C Tseng and S-L Lee, "Design of Wideband Fractional Delay Filters Using Derivative Sampling Method", IEEE Trans. Circuits and Systems I, vol. 57, num. 8, p. 2087-2098, 2010