What's the point of the tank in an air compressor?

Air compressors, including portable air compressors, arrive in at least three varieties:

1. a handheld air compressor, for car tires,
2. a block air compressor, without a tank, still for car tires, but perhaps suitable for continuous usage, and
3. an air compressor with an adjacent air tank, with varying capacities (1 to 5 gallons).

The first two are easy enough to understand. Air comes in from one side, and is compressed in the other.

The practical difference in usage is less clear.

What is the use of the tank in an air compressor fitted with one? If the idea was to fill the tank with compressed air (say up to 120 psi), disconnect the power supply, and take the compressor around to the usage site (whether it's just for tires at 30 psi, or for power tools), then that appears to be a moot point now that air compressors with Lithium-Ion batteries are available. The tank must be needed for something else. Is it just cfm (cubic feet per minute) of air? But the first two kinds also reach 100+ psi, and so they could conceivably be used for the same applications, just not on a continuous basis, and/or with a cooling down period, which is alright for once-in-a-long-while usage.

How does the air circulate? Is it necessary to first fill the tank, or can an air compressor be used directly?