TL;DR Currently:
BRD = Support + DPS
MCH = DPS
Longer Explanation:
If you look at the categories on the sheet, BRD and MCH are “Ranged Physical DPS” they generally do less damage than melee DPS, I assume because they have the advantage of long range attacks and mobility. SE wanted to give people a reason to play a lower DPS job beyond just the range advantage, but not equal the damage output of say BLM and RDM. So they gave BRD a bunch of extra buffs the party could benefit from (not just the TP and MP refresh which I agree hardly gets used now.) MCH on the other hand has diverged and offers no such support outside their cross role abilities. Some are assuming this means that MCH is supposed to be more DPS. However MCH is currently trying to carve out an identity again seeing as how BRD can put out damage with less hassle than MCH and MCH has to use more complex rotations to put out the same damage. The many MCH threads you will see for a while are to help suggest MCH as being THE premier Ranged DPS or offer a different kind of support apart from bard (like becoming the debuffer class instead.)
Support wise, aside from shared role abilities Bard can:
+ 2% Critical Hit rate for the party. (Potentially more damage)
+ 15% Direct Hit rate for the party. (Potentially more damage.)
+ 20% Heal rate to a party member with a song
+ Remove a debuff with a song
+ depending on a song being played can: Increase HP by 15%, Reduce physical Vulnerability by 10%, or reduce magic Vulnerability by 10%
Generally if played correctly having a BRD ups everyone's DPS and can pull some emergency support should a healer feel overwhelmed or get snagged by a paralyze or a silence, all the while barely lowering their personal DPS in the process. So they are a benefit to have.
Support wise, aside from shared role abilities Mechanist can:
+ Lower a targets damage by 10%
+ Knockback a target (but do no damage)
Generally you bring a MCH for more DPS, naturally being ranged means they can put on some constant DPS without having to worry about range restrictions so if played correctly they are a benefit to have for their DPS alone. Their Damage debuff is good for popping before a big boss attack. They don't really compete for the spot BRD has anymore and are more in line with the other DPS roles (but at ranged instead.)