Is there any significant difference between using [[ $a -lt 2 ]] and (( $a < 2 ))?
For example, is one of them faster or more POSIX compliant than the other?
Neither is POSIX-compatible. In a POSIX shell, you can use the command [ "$a" -lt 2 ] or the expression $(( a < 2 )).
In bash, the former is simply the conditional command supporting a superset of the conditional expressions that [ supports, and the latter is a standalone command that exits with status 0 if the enclosed arithmetic expression is non-zero, or 1 otherwise. Other than readability, there is no significant difference between the two when used properly.
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