The usual story of Givens rotations vs Householder reflections is that Householder reflections are better if you want to map a long vector to $e_1$, while Givens is better if you want to map a 2-vector to $e_1$. However, I can't find a reference which explains why Givens is better in the 2-vector case. Can anyone explain?
Typically you don't apply these matrices to a single vector; you apply them to a $m\times n$ matrix, where $n$ is large, so you can amortize the cost of computing the various coefficients. Applying a Householder reflector to a vector (with the formula $x \mapsto x - vv^Tx$) costs $4m-1$. Applying a Givens rotation works only for $m=2$, and costs $6$ operations, which is cheaper.
EDIT: To apply the Householder transformation, you could instead form the $2\times 2$ matrix $Q = I-vv^T$ and apply that to a vector, as suggested by @gTcV in the comments, for 6 operations. But it's not complicated to see that at this point $Q$ is essentially a Givens rotations with two signs swapped, so that's not really anything different from Givens transformations.