Can't draw a diagram here, but I will take a crack.
To start, if you are sailing directly downwind, the faster you go, the lower the apparant wind on your sails. You are out running the wind. So far, so good? The faster you go, the less you have to work with heading downwind.
If you sail upwind, you actually create more wind to work with (increased apparent wind), because you compound the wind as you begin to move into it.
Therefore, it stands to reason that you will get more boat speed sailing into the wind, then away from it. Boy, there are exceptions to that, but let's stay simple.
So why a difference in close hauled and a beam reach? A beam reach remains the lowest point of sail, where you are still being pulled by the lift of the sail, rather than being pushed from behind. Therefore, you may still be creating some increase in apparent wind.
Now the vector part. If you thnk about where the boom is on a close haul vs. a beam reach, this may begin to make sense. Since you are creating a wing with your sail, the boom is much closer to the center of the boat on a close haul, as the wind is coming more directly at our bow. On a beam reach, your boom is eased out so the front of the wing (sail) is facing the wind coming from the side.
For illustrative purposes, imagine that the lift coming off the sail is heading perpendicular to the front of the boom. In a close haul, with the boom in tight, that lift is predominantly pointed off the side of the boat. You don't move sideways, because your keel or centerboard fights the sideways vector, leaving only forward motion. It's like pressing down on marble, but it squirts out from under your thumb. The vector of pressure is toward the table, but the table keeps it from moving in the direction of the pressure and it escapes to the side of least resistance. Your sails are the thumb and your keel is the table.
On a beam reach, the boom is eased out a bit further, so the vector is pointed more forward. The keel fights less, the boat heels less due to less sideward vector and there is more power remaining for forward motion. Sort of like your finger not pressing that marble directly into the table, but at a better angle toward the direction you want it to go.
Hope that wasn't too rudimentary...... or completely confusing.
. Thumbs, marbles and tables, what the heck?