I got carried away and probably covered stuff you already know but there is a simple answer at the bottom if you can't be bothered reading
BW is more specifically Brew Water Weight.
i.e. 15g coffee:250g Brew Water
The Beverage Weight is simply the total weight of your brewed coffee. Which usually sits somewhere between 205-220g with the above ratio (for me).
When using the app you'll find that neither of these variables can be adjusted without affecting the other as the app maintains a specific ratio between their values.
So as you drag one across, the other will change as well.
The Liquid Retention Ratio (LRR) is the value that determines how far apart they are.
It's essentially deciding how much water was retained in the bed of coffee based off the numbers you've given it.
You should see it in a small box next to Moisture and Co2 etc.
This is a number that I adjust frequently so my BW and my Bev values in the app reflect my actual brew.
So when I drag the brew water across to 250g, it may automatically decide my beverage weight is 205g when I actually got 215g out.
By lowering the LRR value I'm letting the app know that less water has been retained in my bed and therefore my beverage weight was larger (215).
This ratio matters because the EXT% is calculated using the TDS% of your beverage weight not your brew water weight.
If you click into Presets, then Settings, you'll find that you can adjust the LRR for both drip and immersion.
Brew Water Weight = how much water went in.
Beverage Weight = how much brewed coffee came out.
LRR = the difference (as a ratio) between BW and Bev.
Hope this helps in some way.