jmc

Both $|x+y|$ and $|x|+|y|$ are nonnegative, so $\frac{|x+y|}{|x|+|y|}$ must be nonnegative. When $x=1$ and $y=-1,$ $$\frac{|x+y|}{|x|+|y|}=\frac{0}{2}=0,$$so this is clearly the minimum.

On the other hand, by the Triangle Inequality, $|x|+|y|\ge |x+y|,$ so $$\frac{|x+y|}{|x|+|y|}\le 1.$$Equality occurs when $x=y,$ so the maximum is 1.

Therefore, $M-m=1-0=\boxed{1}.$