I think what you mean is that a two variable linear equation over the reals describes a line in the Cartesian plane.

Try thinking of a line's slope (as you've described it) as just one measure of that line's "steepness" in the Cartesian plane. Thinking like this, we could use any number of measures to describe "steepness" (for example, angle of inclination provides a measure for the "steepness" of any line)

So it's sort of a philosophical difference. However, there is a a very subtle elegance at work here.... we have this geometric concept of "steepness" which -- very interestingly -- turns out to be related an algebraic concept.

Actually, we can go further. The geometric concept of "steepness" is related to algebra via a simple quotient, which is further related to an analytic concept (rate of change).

It seems so trivial, but a good number of mathematicians make a living by finding small connections between geometry, algebra, analysis, etc.

(Edited)