Why do the spoilers of an Airbus A320 operate so much slower than those on a Boeing 737?

There is a difference in spoiler operation logic between the two aircraft. In the Boeing 737, when the spoilers are armed, you need these conditions for them to extend:

Both thrust levers at idle position.
Compression of one main landing gear (note: compression of any landing gear will deploy flight spoilers. It is the compression of the right main gear that extends ground spoilers ).
- Refer to the picture below.
Radio altitude less than 10 ft.
The detection of main landing gear spin up (more than 60 knots).

If the spoilers were not armed for the landing, they come up when reverse thrust is selected. They can also be manually extended by moving the speed brake lever to the up position.

The spoilers on a B737. The red boxes show the ground spoilers, while the yellow boxes show the flight spoilers.

In the Airbus A320, it is a little bit more complex.

When spoilers are armed or not retracted:

Both main landing gears on the ground.
Both thrust levers at or below idle or reverse selected on one engine with the other thrust lever below MCT.

When Spoilers are in the retracted position but not armed:

Both main landing gears on the ground.
Reverse selected on atleast one engine with the other thrust lever below MCT.

The A320 also has a partial spoiler extension called Phased lift dumping (PLD). The idea of PLD is to accelerate full spoiler extension

When spoilers are armed or not retracted:

One main landing gear on the ground.
Both thrust levers at idle position.

When spoilers are in the retracted position but not armed:

One main landing gear on the ground.
Reverse selected on atleast one engine with the other thrust lever below MCT.

Note: In the A320, for spoiler extension, the wheel spin up speed should be greater than 72 knots.

Now that how both spoiler systems work is understood let us see how the spoiler deflection is delayed in the A320 when compared to the Boeing 737. First of all, in the B737, the spoiler extension (flight spoilers) require only the compression of one main landing gear. But in the A320 the compression of two main landing gears is required for full spoiler extension. In the A320, if one wheel is detected on the ground, the spoilers are only halfway extended, because of previously discussed PLD. When the other gear is compressed the full extension occurs. In very rare circumstances will both landing gear compression will be detected at the same time. There will always be a delay. So, the delay you see in the A320 spoiler extension is the phased lift dumping blending in with the full spoiler extension.

The other reason for the delay is the wheel spin up speed that is needed for spoiler extension. In the B737, a speed of 60 knots is required, while in the A320, a wheel spin speed of 72 knots is required for spoiler extension. So, the speed of deployment is over 12 knots higher in the A320, compared to the B737. This is the other reason why the spoiler deployment is slower in the A320.