Answer
$2-Bromo-2-methylbutane$ will react faster by the $S_N1$ mechanism.
Work Step by Step
We know that the structure which forms a more stable carbocation will react faster by the $S_N1$ mechanism. In the given pair, $2-Bromo-2-methylbutane$ will form a $3^\circ$ carbocation, which is more stable. So, $2-Bromo-2-methylbutane$ will react faster by the $S_N1$ mechanism.