units - What are some applicable problems with the correct usage of G?

Usually, using different units is done for convienience, as HDE 226868's answer explains. If we're talking about stars, then measuring masses relative to the Sun is sensible. The Sun is the closest star to us, and astronomers understand it much better than any other, making it the de facto standard of comparison. Similarly, on the interstellar scales the parsec is more convenient than the meter. The conversion is trivial and doesn't change anything physically. Recall $F = ma$ to see how the newton corresponds to base SI units:
$$mathrm{N}cdotleft(frac{mathrm{m}}{mathrm{kg}}right)^2 = left(frac{mathrm{kg}cdotmathrm{m}}{mathrm{s}^2}right)frac{mathrm{m}^2}{mathrm{kg}^2} = frac{mathrm{m}^3}{mathrm{kg}cdotmathrm{s}^2}text{,}$$
so we can just multiply the SI value of $G$ by
$$left(frac{1.989times 10^{30},mathrm{kg}}{1,mathrm{M}_odot}right)
left(frac{1,mathrm{pc}}{3.086times 10^{16},mathrm{m}}right)
left(frac{1,mathrm{km}}{1000,mathrm{m}}right)^2text{.}$$
But there could be another reason to use solar masses in particular. The value of Newton's gravitational constant is:
$$begin{eqnarray*}
G &=& 6.674times 10^{-11},frac{mathrm{m}^3}{mathrm{kg}cdotmathrm{s}^2}\
&=& 2.959122083times 10^{-4},frac{mathrm{AU}^3}{mathrm{M}_odotcdotmathrm{day}^2}\
% &=& 3.964015993times 10^{-14},frac{mathrm{AU}^3}{mathrm{M}_odotcdotmathrm{s}^2}\
&=& 4.300917271times 10^{-3},frac{mathrm{pc}}{mathrm{M}_odot}frac{mathrm{km}^2}{mathrm{s}^2}text{.}
end{eqnarray*}$$
The difference in precision is real. The 2010 CODATA value of $G$ is $
(6.67384pm0.00080)times 10^{-11}$ in the SI units, which has a relative uncertainty of $10^{-4}$, so we can't expect more than about $4$ significant figures. On the other hand, the standard gravitational parameter $mu = GM$ of the Sun is known to a much greater precision than either $G$ or their masses individually, so expressing $G$ in solar masses gives a more precise value. The basic reason for that is simple: it's much easier to compare the orbits of planets to each other than, say, to compare the Sun to the International Prototype of the Kilogram kept in a French vault.

Of course, masses of other stars and interstellar distances tend to have uncertainties larger than that, so this is irrelevant on that scale. Still, within the solar system, using units of solar masses (or just $GM_odot$ directly) allows one to make much more precise measurements, and similarly Earth masses for satellite orbits around the Earth.

Measuring $G$ in terms of parsecs (or light-years) and $(mathrm{km}/mathrm{s})^2$ would be more convenient any time one is dealing with galactic-scale distances, such as in the galactic rotation curve:

(Image from Wikipedia).