You may get a more in depth answer from r/AskEngineers but I can try to help. I have no knowledge of the history of sailing or ships myself but am a civil engineer and can help explain the reason and science behind why the thin sails would be preferred.

Having a series of smaller, stacked sails is a technological and engineering advantage to having one huge sail that you often see on Medieval sailing ships like a Cog. As technologies and understanding of ship building, textiles, and navigational science increased this would lead to more efficient and powerful sails.

Some specific reasons that smaller, stacked sails are more practical for sailing: When a sail is 'catching wind', the wind is 'pushing' on the sail, creating force. Just like wind moves the fins of a wind turbine. The larger the sail, the more surface area that is 'catching' wind and the larger the force that is being applied to that single sail. To keep the sail, rigging, mast and rest of the ship from being damaged from that force you would need to increase the size and strength of the parts. A big, beefy mast... strong, heavy duty fabric sail... thick, beefy ropes. All of that increases the overall weight making the ship heavier. Heavy ships are slower that lighter ships. It's the same core dilemma rocket scientists face when designing rocket ships! Faster ships are great when you have global, commercial empires!

So, having a series of smaller sails means that each individual sail is not receiving as much force and stress. So individually each sail (the fabric itself) can be lighter. The rigging, rope, mast, and everything else can also be less beefy resulting in a lighter, faster ship. But each sail working together adds the forces together to push the ship with the same force as the single, huge sail.

Another advantage of having multiple sails is that you can deploy specific sails. If the wind is blowing faster near the top of the sail, then the ship could keep those sails full while other sails that are not receiving any wind can be 'rolled up' (excuse my lack of nautical vocabulary). This also helps the speed of a ship. Say a ship has just the one giant-sail and the wind is only 'pushing' through the top 25% of the sail. The other 75% of the sail is acting like a parachute break, slowing the ship down! The bottom 75% is trying to 'cut' through the air, creating large amounts of drag. Now back to our multi-sail ship. We could leave those top sails open to catch the wind but close the bottom sails, reducing the surface area of the unnecessary sails and create a lot less drag.

Smaller, stacked sails would also help with repair time. Back to our one giant-sail ship. If you are sailing that across the Atlantic and your sail is damaged you would have to take the whole sail down. During the repair time the ship is just a floating barge making very little to no progress. With multiple sails, when a sail is damaged it could be repaired while the others remain operational.