Except for code which does a significant number of floating-point operations on data that are held in cache, most floating-point intensive code is performance limited by memory bandwidth and cache capacity rather than by flops.
$v$ and the products $Av$ and $Bv$ are all vectors of length 2000 (16K bytes in double precision), which will easily fit into a level 1 cache. The matrices $A$ and $B$ are 2000 by 2000 or about 32 megabytes in size. Your level 3 cache might be large enough to store one of these matrices if you've got a really good processor.
Computing $Av$ requires reading 32 megabytes (for $A$) in from memory, reading in 16K bytes (for $v$) storing intermediate results in the L1 cache and eventually writing 16K bytes out to memory. Multiplying $Bv$ takes the same amount of work. Adding the two intermediate results to get the final result requires a trivial amount of work. That's a total of roughly 64 megabytes of reads and an insignificant number of writes.
Computing $(A+B)$ requires reading 32 megabytes (for A) plus 32 megabytes (for B) from memory and writing 32 megabytes (for A+B) out. Then you have to do a single matrix-vector multiplication as above which involves reading 32 megabytes from memory (if you've got a big L3 cache, then perhaps this 32 megabytes is in that L3 cache.) That's a total of 96 megabytes of reads and 32 megabytes of writes.
Thus there's twice as much memory traffic involved in computing this as $(A+B)v$ instead of $Av+Bv$.
Note that if you have to do many of these multiplications with different vectors $v$ but the same $A$ and $B$, then it will become more efficient to compute $A+B$ once and reuse that matrix for the matrix-vector multiplications.