In our experience, there is a huge variety of temperaments and drives within the individuals of both breeds.
Supposedly borzoi are reserved, quiet, and laid back. For the most part that is true of Strider, except for the reserved part. He is extremely extroverted and loves to be the center of attention, and doesn't care who it is coming from. Just within his litter, there was a huge range of temperaments. He has a bro that was VERY active and outgoing, more like a border collie in a borzoi body, there was one brother that just laid on the ground like a pancake and refused to get up and play, or move, or anything. He just liked to sleep, and stretch. Strider is in-between those two. He also has another brother who hates meeting new people, likes to have his own space, hates to leave the house, etc. There were so many personality and energy level extremes in that litter it is hard to say what you are going to end up with when you look at the breed as a whole. Strider and his border collie wannabe brother are both extremely handler oriented, very food, toy and praise motivated, thus easy to train. I have also met borzoi that were just kind of indifferent to everyone around except their one human, and only when that one human had something they wanted such as a favorite toy, or a leash.
The same thing seems to apply to greyhounds. I think Sismorphine said once that there is a greyhound out there to fit every person, and I think that is true. When we were looking into adopting one and started meeting dogs we found a huge variety of personalities, energy levels, and drives.