There is some redundancy in the solution. Referring to JS:

if (target_diff === 0) { // we've found a triplet with an exact sum return targetSum - target_diff; // return sum of all the numbers }

can just be

if (target_diff === 0) { return targetSum; }

This one is in all solutions.

And the following is only in the JS:

if (Math.abs(target_diff) < Math.abs(smallest_difference)) { smallest_difference = target_diff; // save the closest difference }

is redundant, because it's also in the next if conditional with the exact same result.