Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?

Question

I would like to know what really are the main differences (in terms of "usefulness") among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula.

Could you provide examples of situations where one form "works better" than another?

Also, what are the actual benefits of the new generalizations proposed for example in the following articles?

1. BLUMENTHAL, L. M., Concerning the Remainder Term in Taylor's Formula. Amer. Math. Monthly 33, pp. 424-426, 1926.

2. BEESACK, P. R., A General Form of the Remainder in Taylor's Theorem. Amer. Math. Monthly 73, pp. 64-67, 1966