"[ ]" vs. "[[ ]]" in Bash shell

test's string equality operator doesn't do globs.

$ [ abc = *bc ] ; echo $?
1
$ [[ abc = *bc ]] ; echo $?
0

[[ is a bash built-in, and cannot be used in a #!/bin/sh script. You'll want to read the Conditional Commands section of the bash manual to learn the capabilities of [[. The major benefits that spring to mind:

• == and != perform pattern matching, so the right-hand side can be a glob pattern
• =~ performs regular expression matching. Captured groups are stored in the BASH_REMATCH array.
• boolean operators && and ||
• parenthèses for grouping of expressions.
• no word splitting, so it's not strictly necessary to quote your variables.

The major drawback: your script is now bash-specific.

Also - under what circumstances should [ ] be used vs. [[ ]] and vice versa?

It depends. If you care about portability and want your shell scripts to run on a variety of shells, then you should never use [[. If you want the features provided by [[ on some shells, you should use [[ when you want those features. Personally, I never use [[ because portability is important to me.