Here's why I bought the Woo WES over the BHSE. First, I'm an engineer and an (amateur) audiophile. I certainly don't know as much about headphones and high-end audio as many people here -- especially Spritzer, who's somewhat (over-?)revered as a god on this site.
Anyway, I wanted the best sound for a sizeable investment that would last as long as possible.
If you read the reviews (e.g., Tyll's at Inner Fidelity and AV Guide) critically, there isn't much difference among high-end amps. They both talk about how amps offer the least difference among components and how "hair-splitting" the comparisons are. They also refer to advantages that the WES has specifically over the BHSE, such as tighter control of bass. Clarity and detail retrieval are also above any other amps.
I am basing this on expert reviewers' impressions. If some of you want to challenge these, so be it, but these reviews are as objective and professional as possible.
Another advantage, the Woo WES is easier to upgrade over time. You can get the base version and add better tubes later. I'm not sure if you can upgrade the volume control of the BHSE so easily down the line and how much it will cost.
Finally, I like the fact that the WES has no circuit boards, no ICs, and uses point-to-point wiring. Every integrated circuit will one day fail. Every printed circuit board will one day get a short or have a component failure. What happens with the BHSE in 20 years when that happens? Will Justin be around? Hopefully... but what if he isn't? The BHSE will become an expensive, huge, paper weight. With the WES, you just replace the failed component.
I'm not really a fan of the Woo company, believe it or not. Despite this, I still think that if you favor longevity, low (relatively) entry price, and top-tier sound, the WES is the one to get.