In the Wikipedia entry for Waring's problem, the section on $G(k)$ starts as: “From the work of Hardy and Littlewood, more fundamental than $g(k)$ turned out to be $G(k)$, which is defined...” There is no real justification or citations.
Do you believe this? Is there a specific, objective sense in which $G(k)$ is more fundamental? My answer is that the Hardy-Littlewood method itself works only for sufficiently large $x$ (let's say you are interested in density of subsets of $\mathbf N$ in the interval $[1;x]$), so it's better suited to handle $G(k)$ than $g(k)$. Is there a better answer?