What is the difference between live stake and active stake?
Staking with a pool implies the benefit of earning rewards on that amount of ADA you delegate to them to mine those rewards on your behalf.
The question regarding the difference between live and active stake pertains to the amount of stake living at two different points of time.
"Live" stake is the stake living in the pool right now. That is, it is the amount of ADA being delegated in real-time. This stake amount can vary at any time. If somebody delegates 100 ADA the live stake monitor will reflect the addition of the 100 ADA immediately.
"Active" stake, on the other hand, is the amount of stake that was measured at a point of time (via what is called a "snapshot"). Active stake is measured at the moment of transition between epochs.
Epochs are the five-day intervals for pool mining operations to have their stake registered, tabulated, and calculated.
Active stake remains fixed for five days and is the stake amount used to qualify the pool to mine rewards during an epoch.
The proposition for delegators to understand is that the Cardano network needs an account of how much ADA is being delegated, where it is being delegated, and so on, to be able to attribute rewards. This accounting of the amount of ADA being delegated cannot be done on a running basis. It has to be measured at a moment in time to function properly. A dynamic running system of every movement of ADA at each and every moment is not feasible. So, the decision was made to measure the account of all ADA staked each period of 5 days.
This answer was published on 20 January 2021
Further resources to discover more on this theme:
Reddit post: Still don't understand what is active and live stake.
YouTube post: Live Stake and Active Stake, an Important Difference for Stakers on Cardano