In these cases it's a good idea to look up the precise definition and usage of these functions in the respective documentation pages. Often there are differences between system, as even in math books there's no
one true convention that everyone follows.
The Mathematica documentation is very good in that it will give a precise definition of every mathematical functin of distribution (usually in a very clear and understandable manner). In my experience the documentation of R packages is usually not nearly as good: it will often just say "this is a gamma distribution" or "this is an incidence matrix" without clarifying which meaning of those words it's referring to.
So you can
look up that Mathematica's GammaDistribution[a,b] represents a dsitrbution with PDF proportional to Exp[-x/b] x^(a-1).
It seems R has several packages and several functions for gamma distritbuions.
The set of R functions named as *gamma() have two settable parameters: the shape and scale. "Shape" corresponds to 'a' in the above formula and "scale" corresponds to 'b'.
As David said, Mean@GammaDistribution[a,b] is a*b.
I'm not very fluent in R, but if I do
x <- rgamma(1000, shape=2, scale=3)
mean(x)
I get a value close to 6=2*3, i.e. consistent with Mathematica.
Conclusion: you'll need to look up the precise meaning of the parameters the books uses and see how they transform to the parameters Mathematica uses. Most likely the second parameter in the book refers to "1/scale", which R calls "rate".