Gibbs Free Energy Notation
$G$ is the Gibbs free energy of a system. It is a conceptual quantity in the sense that there is no reference point that defines $G = 0$ for a substance (unlike entropy). Whenever you see a plot of $G$, notice that the scale is not labeled with absolute values (no indication of where the origin is).
$\Delta G$ is the change in Gibbs free energy, i.e. $G_\text{final} - G_\text{initial}$. Because it is a difference, there is no issue with the lack of reference point.
$\Delta_\mathrm r G$ is the change in $G$ as a reaction proceeds, i.e. $\Delta_\mathrm r G = \frac{\mathrm dG}{\mathrm d\xi}$ with $\xi$ the extent of reaction variable. Because of this definition, it is an intensive quantity with dimensions energy per amount of substance. It is called the Gibbs energy of reaction.
$\Delta_\mathrm r G^\circ$ is the standard Gibbs energy of reaction, for the special case when all reactants and products are at standard state.
$\Delta G_\mathrm f^\circ$ is the standard Gibbs energy of formation of a substance, the $\Delta_\mathrm r G^\circ$ for the reaction that makes the substance from the elements (where all products and reactants are at standard state).
$\Delta G^\circ$ is for any process at standard state (e.g. ice melting at a pressure of 1 bar). It is also sometimes used instead of $\Delta_\mathrm r G^\circ$, but that is confusing because the meaning and the dimensions are different.



  I've seen the first two used interchangably, and seen $G$ specifically referred to as change in Gibbs Free energy. Is this common? Are there cases where $G$ refers to absolute Gibbs Free Energy?


$G$ should always refer to absolute Gibbs energy, and never to a change in Gibbs energy.


  For those with the circle ($^\circ$), I've seen that referred to as "standard", and this means that they are in reference to standard pressure or concentration. Does this make their use noticably different?


Yes, because the Gibbs energy is concentration-dependent. With $Q$ as the reaction quotient,

$$\Delta_\mathrm r G = \Delta_\mathrm r G^\circ + R T \ln Q$$

As concentrations change in a reaction, $\Delta_\mathrm r G$ changes but $\Delta_\mathrm r G^\circ$ does not. When a reaction approaches equilibrium, $\Delta_r G$ approaches zero (but $\Delta_\mathrm r G^\circ$ doesn't change, as just mentioned).

The reason that $\Delta_\mathrm r G$ is concentration-dependent is that the Gibbs energy has an entropy component ($G = H - T S$), and there is an entropy of mixing that depends on concentrations (the $R \ln Q$ part).