Relative to each other...
IME, Dex monks shine when:
1) There is significant ranged melee and/or facing foes with spells/powers/abilities forcing Ref saves or targeting touch AC, like a Beholder (all of their eye effects are rays, IOW, Ranged Touch attacks).
2) When they get full use of their AoOs and they have to weather a lot of attacks, namely when they are facing a large number of foes, or to a lesser extent, against a foe who has a large number of attacks.
3) Acting as substitutes for or as stealthy muscle support to a Rogue (or similar PC) on an advance/"scouting" mission- they're that much more likely to succeed on a critical Move Silently roll that could mean the difference between a bloodless scouting mission and a trip to the dungeons...or the graveyard.
IME, Str Monks shine when:
1) You must take down or harry a single elusive or "hardened" target, like a heavily armored spellcaster on the back rows or dragon in his lair.
2) the BBEG's mooks & minions are relatively few in number but high in AC & HP- extra AoOs for high Dex make much less of a difference.
3) Books used in a campaign are limited- there are far more feats, etc. that favor a Str based combatant over a Dex one in the Core & Completes. Even special manuvers that Monks should usually use, like Grapples, Trips & Disarms mainly favor a strong combatant.
Several of the Feats that favor Dex builds are in campaign-specific sources (like Serpent Strike from Eberron which makes the Longspear a Monk weapon), sources not widely used (Serpent Fang from Sandstorm) or in "controversial" rulesets that aren't universally allowed (Stand Still from the XPH, or Pole Fighter, Ring the Golden Bell and Unorthodox Flurry from DCv1).