Slope: rise over run, for any straight line. Example: y=mx + b
Constant of proportionality: a constant by which you multiply your independent variable (x) to get a specific value of the dependent variable (y). Example: y=kx.
The main difference here is that y=mx+b may or may not pass through the origin, whereas y=kx always passes through the origin.
Both k and m are "slopes," with k pertaining to direct variation y=kx, and m pertaining to slope of a line which may or may not pass through the origin.
In this limited sense, both k and x are slopes. k could be the slope of a straight line passing thru the origin, same as m, but m could not be the constant of proportionality if the line in question does not pass through the origin.