Some differences:
Clear[x, y];
f[] := x;
Assuming[x == 5 && y == 15,
{x, y, Simplify[f[]], Simplify[x], Simplify[y], Simplify[y^2]}]
(* {x, y, 5, 5, y, 225} *)
With[{x = 5, y = 15},
{x, y, Simplify[f[]], Hold[x], Hold[y], Hold[y^2]}]
(* {5, 15, x, Hold[5], Hold[15], Hold[15^2]} *)
The only real effect of Assuming[P, expr] is to change $Assumptions temporarily to $Assumptions = And[P, $Assumptions]. This might or might not affect the computation of expr, since very few commands use $Assumptions. Simplify[] is one of the few. But since it does not consider 15 to be "simpler" than y, it leaves y unchanged. But 5 is simpler than x and 225 is simpler than y^2. Maybe someday they will have a ChatSimplify[] so that your personal AI can advise Simplify[] on what you consider simpler. If you are ever looking for a function that tries to apply assumptions without worrying about "simpler," I'd suggest Refine[].
With[] replaces all the literal occurrences of x and y in the code following the list. Note that the code f[] does not contain a literal occurrence of x. When it is evaluated and becomes x, With[] has finished with its job and the value of f[] is not replaced by 5.
Maybe you were thinking the syntax of assuming is Assuming[{x = 5, y = 15}, expr]? The equals should be double equals. While you may give a list of conditions as the first argument, Assuming[] converts the List[] to an And[] when it combines the conditions with $Assumptions.
