What is the difference between using bash and sh to run a script?

That depends on the system you are running. One many OSes, especially Linux based ones, sh is a link to bash.

In such case, there are still some differences in behavior where bash try to be more like traditional bourne shell when called sh, but it still accepts most bashisms.

On some other OSes, like Debian based ones, sh is provided by dash, not bash. That makes a much bigger difference as dash doesn't support bashisms, being designed to be a clean POSIX shell implementation.

On proprietary OSes, sh is often either provided by a POSIX compliant ksh88 which like dash, doesn't implement bashisms. On Solaris 10 and older, depending on what your PATH is, sh will likely be a legacy Bourne shell, predating POSIX.

In any case, you likely got the same output with your test simply because your script was not using any bash specific command, option or syntax.

When you run ./executable, what shell will be run essentially depends on the shebang written at the beginning of the .executable script. That will be bash if the shebang specifies it:

#!/bin/bash
....

If there is no shebang and you call the script from a POSIX compliant shell, the script should technically be executed by the first sh found in the PATH. Many shell interpreters like bash, dash and ksh are considering themselves to be POSIX so will interpret the script. Note that the SHELL environment variable is not used here.

Yes it could be, but it does not have to be. The two shells are not the same, but the basics are compatible. Check this post for more details.

You can use the following line in a script to check the interpreter that is used:

ps h -p $$ -o args='' | cut -f1 -d' '

You can explicitly specify the shell that shall be used when the script is executed directly with the sheebang line:

#!/bin/sh

or

#!/bin/bash