I don't know how to word this mathematically accurate, but the general idea is the following:
Let's try to dissect the half A to match it with B. Since the number of pieces is finite, the borders of B will be composed of a finite number of pieces, all of which will be straight lines.
Now, if we dissect the part B, all each straight cut (even straight parts of a more complex cut) will produce two straight borders for the new pieces, they will be of equal length but of opposite direction (the direction of a straight border part is a vector that's perpendicular to it and points outwards). So the total (added for all pieces) balance of straight borders directions (e.g. 1 cm "north" border, 1 cm "west" border and sqrt(2) "southeast" border) will not change no matter how we cut. When we join the pieces together, it won't change also since these borders will negate each other when joining.
So, with such operations we can only create a figure with the same border balance (opposite directions negate each other when calculating) as the initial figure. But the other half of a square has completely different stats: 1 cm "south" (or -1 "north" for comparison, since these directions are opposite), 1 cm "east" (-1 "west) and sqrt(2) "northwest" (-sqrt2 "southeast"). So we cannot turn one half into another.
The wording is far from being perfect, but I hope that the idea is clear.