If I evaluate brf[AAA, WWW] there are no instances of AAA or WWW. This is because the definition,
brf[a_, w_] := ArgMax[{v, e >= 0, e <= 1, Uem > Uun}, e];
is a "literal" rewrite rule. It replaces the instances of a and w that literally appear in the (unevaluated) text ArgMax[{v, e >= 0, e <= 1, Uem > Uun}, e]. However there are no symbols a or w present before it is evaluated. The a and w appear only when the RHS is evaluated, which happens after the literal replacement has finished. But I second Gianluca Gorni's suggestion that the plot looks like there were numerical issues.
If you wish not to write the RHS with a and w present, you could try to define brf as follows:
brf[a0_, w0_] :=
Block[{a = a0, w = w0}, ArgMax[{v, e >= 0, e <= 1, Uem > Uun}, e]];
Note that Plot[] and Table[] effectively use Block[] to give values to w and a respectively, so the functions almost work inside those commands. In the first plot, brf[w] is evaluated when w has been given a value by Plot[]; so it works properly, more or less. I get an error from NMaximize which you don't mention. In the second plot, brf[a, w] is evaluate when Table[] has given a the value 0.9 but w has no value. The result is then evaluated in Plot[] after w has been given a value. Maybe this works, maybe not; it may explain the difference. I couldn't track it down.
I would define brf[] as follows:
brf // ClearAll;
brf[a0_?NumericQ, w0_?NumericQ] :=
Block[{a = a0, w = w0}, ArgMax[{v, e >= 0, e <= 1, Uem > Uun}, e]];
Or I would make the dependence of Uem on w etc. explicit (define it to be a function Uem[w,...]).
See https://support.wolfram.com/12502 for a discussion. I don't know if the above fixes the plot because Plot[] ran for several minutes, longer than it took to type this up. I wasn't sure when it would finish and had to move on to other things.
