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 costs $4m-1$. Applying a Givens rotation works only for $m=2$, and costs $6$ operations, which is cheaper.