Why are there two different SI units of magnetic charge? Taking a look at the wiki on magnetic charge, there are two different forms of the Dirac Quantization Condition in SI units: the Weber form 
$$\frac{q_e q_m}{2 \pi \hbar} \in \mathbb{Z}$$
and the Ampere*Meter form 
$$\frac{q_e q_m}{2\pi \epsilon_0 \hbar c^2} \in \mathbb{Z}$$
Can someone please elucidate why the SI system would provide two possible values of the magnetic charge? Unless I am missing something, they are numerically different, as $\epsilon_0 c^2 \neq 1$. I have had trouble finding sources on this topic. Which one would be used in the Coulomb force equation for magnetic monopoles, and under which circumstances?
 A: The two equations are not both in SI. The upper equation is in CGS units, a system closely related to SI that was a favorite of spectroscopists because of the simple form of Coulomb's law in it. That lead to it being the unit system of choice in standard texts, Purcell and Jackson, which were updated only recently, and in astronomy. Some theorists also like CGS because in it electric and magnetic fields have the same units.
I'm not, personally, a fan of CGS for two reasons. First, $\mathbf{E}$ and $\mathbf{B}$ should only have the same units when the speed of light, $c$, is 1. Just like the units for energy versus momentum. Second, the $4\pi$ belongs in Coulomb's law because it gives the denominator physical meaning. With $$F = \frac{q_1 q_2}{\epsilon_0 4\pi r^2}$$ it is clear that the denominator is the area of a sphere, as it should be. When the $4\pi$ is moved to the field equations, they just seem arbitrary.
Here's a wiki page on E&M equations in different unit systems.
A: The 'Weber' interpretation is due to Kennelly, comes from $F=Q_mQ_m/4\pi\mu r^2$, symmetrically with Coulomb's Law.  While there is no magnetic charge, the terminology is balanced if one supposes that F = HQ, and B is defined as D.
The 'Ampere-metre' version is due to Sommerfield (of 'fine structure constant' fame).  This can be rewritten as C.(m/s), such as in Qv.  An electric current is presented as Qv/l, = C.(m/s)/m = C/s = A.
Both interpretations are to be found in various texts.  Other variations are things like m=µIA (Wb.m) or m=IA (Am²).  Since this is moment of magnetism, the unit of magnetism is m/length = Wb, or A.m.  
The A.m system gives numbers closer to unity than the Weber system, which is why it has some currency.
A: There may be other versions of magnetic charge in SI units, for example, coulumb (same unit as electric charge), where Dirac Quantization Condition is:
$$\frac{q_eq_m}{2\pi\hbar\mu_0c}\in\mathbb{Z}$$
Actually it depends on which form of Lorentz-Force you'd like to use. Lorentz-Force is important to define magnetic B field. (That's the reason why it has different forms between SI and Gaussian.) Now it is also important if magnetic charge introduced.
The 'weber' version uses this:
$$\mathbf{F}=q_e\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)+q_m\left(\dfrac{\mathbf{B}}{\mu_0}-\varepsilon_0\mathbf{v}\times\mathbf{E}\right)$$
The 'ampere-meter' version uses this:
$$\mathbf{F}=q_e\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)+q_m\left(\mathbf{B}-\dfrac{1}{c^2}\mathbf{v}\times\mathbf{E}\right)$$
The 'coulomb' version uses this:
$$\mathbf{F}=q_e\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)+q_m\left(c\mathbf{B}-\dfrac{1}c\mathbf{v}\times\mathbf{E}\right)$$
See my note for more detail.
Unlike CGS unit systems (where length-mass-time are enough), SI unit system defines the forth dimension (electric current) for electromagnetism. If SI likes to define the fifth dimension for magnetic charge or magnetic current, there will be another new version.
