What is the relative strength of the weak force? When comparing the weak gauge coupling to the electromagnetic gauge coupling (fine structure constant alpha) textbooks give the value ("relative strength") 10^-5 or the value 10^-13. Why are the values different?
 A: Well, there is a mismatch of what you are comparing, and what you think you can do with the "answer". The first α is really  silly (see footnote). A popular text by Griffiths gives a ratio of $10^{-11}$ with the warning: "The “strength” of a force is an intrinsically ambiguous notion- after all, it depends on the nature of the source and on how far away you are. So the numbers in this table should not be taken too literally, and (especially in the case of the weak force) you will see quite different figures quoted elsewhere.".  Others go for $10^{-13}$. Caveat scholar!
If you compare potentials, the infinite-range EM potential goes like 1/r,  whereas the very short range weak "potential" goes like $e^{-mr}/r$, instead, where m is the mass of the W or the Z, so the potential cuts off at $r\sim 1/m$, close to 0.003 fermis, well within the confinement radius of a hadron (1 f) where  processes such as β-decay happen.
The actual gauge couplings themselves are comparable, of course, since they came from the same  underlying mechanism, and, if anything, the charged current coupling g is slightly larger than the EM coupling!
Most books compare amplitudes-squared for some characteristic distance r, or energy (its inverse), and extract a ratio of effective couplings, or squares of couplings (α, involved in decay widths), to give you an impression of the ratio of electromagnetic versus weak cross-sections or inverse lifetimes; so,  to give you a glimpse of what weak suppression to expect. E.g., in nuclear units of a fermi, a W-propagator weak amplitude squared amounts to a suppression of  $1/m^4\sim 10^{-12}$. But if you look at total cross sections at 10 GeV,
the suppression dwindles to just $10^{-4}$, since EM cross sections decrease like 1/s, while weak ones increase like s.
The thing your book should do right away, instead of formulating blind "theological" rules which cannot be made elegant, is to steer your attention to

*

*The EM decay lifetime of the $\tau(\pi^0)\sim 10^{-16}s$; contrasting it to the weak decay $\tau(\pi^\pm)\sim 10^{-8}s$. So, here, very crudely, EM is "stronger" by 8 orders of magnitude than the weak force.


Ν.Β. (Geeky) Unwisely & unfortunately, a ratio of $10^{-5}$ for α s has wended its way into popular literature, based on grimly unfortunate comparison of generic strong decays to weak purely hadronic decays of hyperons. This involves poorly understood complications due to long distance strong physics and is a notoriously unreliable estimate, based, as it is, on a muddy "paradox". By far the least reliable handle on the weak interactions.
A: The coupling constants for interactions can only be usefully compared in terms of dimensionless coupling constants. The size of a dimensioned constant depends on which system of units you use. For instance, I drive at 60 in the US and 100 in Europe.  Do I drive faster in Europe? The funny numbers you show differ in their dimension or in calling
alpha$/M_p^2$ the weak coupling constant G.
The GSW theory of electro-weak interactions requires the Weak gauge coupling to equal the electromagnetic gauge coupling (fine structure constant alpha).  The 'weak' in the weak interaction is because it enters nuclear physics as alpha times $M_p^2/M_W^2\sim 10^{-5}$.  What also makes the coupling constants different is that alpha is a function of the four-momentum transfer squared $(Q^2)$.  This makes alpha about 0.6 applied to baryon structure, but about 0.2 in the region of the Z gauge boson mass.
