Original

A simple example demostrating this effect looks like this:

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pragma solidity ^0.4.19;

contract Test {
  function test(uint[20] a) public returns (uint) {
    return a[10] * 2;
  }

  function test2(uint[20] a) external returnss (uint) {
    return a[10] * 2;
  }
}

Calling each function, the public function uses 496 gas, while the external function uses 261 gas.

The difference is because in public functions, Solidity immediately copies array argument to memory, while external functions can read directly from calldata. Memory allocation is expensive, whereas reading from calldata is cheap.

The reason that public functions need to write all the arguments to memory is that public functions may be called internally, which is actually an entirely different process than external calls.

Internal calls are executed via jumps in the code, and array arguments are passed internally by pointers to memory. Thus, when the compiler generates the code for an internal function, that function expects its arguments to be located in memory.

For external functions, the compiler doesn’t need to allow internal calls, and so it allows arguments to be read directly from calldata, saving the copying step.

As for best practices, you should use external fi you expect that the function will only ever be called externally, and use public if you need to call the function internally. It almost never makes sense to use the this.f() pattern, as this requires a real CALL to be executed, which is expensive. Also, passing arrays via this method would be far more expensive than passing them internally.