I voted 'moderately important' but my real answer is probably 'other'. I feel very strongly that balanced mechanics are essential to prevent the mechanics of the game from distorting the story, but on the other hand I also feel that to a certain extent what I consider 'balanced' is admittedly skewed by my own biases about what is cool. In other words, even if it allows you to do something really powerful, if I think that its cool and that I can handle the consequences of player power, then its in.
So for instance...
Paladins automatically saving vs. fear... OUT as full immunities are almost always to powerful IMO.
Being able to detect evil, alignment, or lies with complete certainty... OUT as it makes it too easy to short cut complex storylines.
Fly spell... WEAKENED (fourth level spell) as ubequitous flight makes interesting dungeon design too hard.
Prestige classes... OUT as the violated the spirit of the game design, led to twinkism, and were too diverse to keep up with.
Various official shield feats... OUT because it seemed unflavorful to just treat being good with a shield as the ability to use it as a weapon.
Spells/Items that give big boosts to skills... OUT or WEAKENED because they end up making skills irrelevant.
Multishot and other powerful ranged feats... OUT because I've had problems with archers just seeming to be the answer to every tactical problem I present the player's.
On the other hand...
Additional bonus feats for many classes... IN because IMO cleric and rogue were too strong relative to the other classes especially at low levels, fighter had few reasons to go to high levels, and the other combat classes seemed to pale compared to fighters (people just splashed them to gain the front end benefits).
More powerful combat feats for high level fighters... IN because fighters seemed to peak at around 12th or so.
More powerful arcane feats for evokers, abjurists, and summoners... IN because those archaetypes seemed to suffer under 3rd edition (evokers because monsters had more hit points, more resistances, and evasion, abjurists because the concept was neglected, and summoners becauses the monster summoning spells were really weak on thier own) compared to enchanters, necromancers, and transmuters.
And in some cases....
Monks... OUT not because they are strong (they aren't) or because they are weak (though they are) but because they just don't fit.
Druids... OUT (ditto) and Shamans (Green Ronin) in.
Improved Unarmed strike and the associated feat trees... WEAKENED because they violate my suspension of disbelief as a swordsman. I don't care if you are Bruce Lee, the disadvantage you have versus an armed opponent of reasonable competance is enormous.
Much of what I allow would strike most DM's I'm sure as either exceptionally harsh or too generous depending on which ruling we looked at, and depending on what they were doing they'd probably be right. But my point is that balance very much depends on what you are trying to accomplish. In a hack and slash game, detect lie is probably too weak to be worth considering. In an intrigue game were everyone is hiding something, detect evil, detect lie, and such is overpowered.