Rogue to monk
For PvE, you are able to match the DPS of a Rogue, however you won't be able to do so until later gear levels and once you have Rune of Reorigination. At 502 ilvl, Rogue will pull more, but neat 520 you both start to even out. The rotation is more complicated however, more akin to a Ret Paladin or Enhancement Shaman's rotation but not as complicated as, say, an Unholy DK.
For PvP, Monk plays much more like an extremely mobile DK than a Rogue. You have about half as much control as you do on a Rogue but between Fists of Fury, Leg Sweep/Ring of Peace, Paralysis and Grapple Weapon, you can still pretty easily take control of a fight. Instead of evaporating as soon as someone catches you in a stun or fear and your trinket is down, you pretty much tank through the damage as a Monk with excellent self healing and Touch of Karma, which is on a pretty low CD. Fortifying Brew is a pretty amazing CD as well but I don't hit it often, at least not compared to how often I hit Cloak or Evasion on my Rogue.
Once you nail it down, you'll be practically unkillable in 1v1, easily shutting down burst from all classes, including Rogue. (Using Diffuse Magic or Touch of Karma when a Rogue uses Shadow Blades is pretty devastating.) You'll be really strong in 2v2 as well, especially with a Disc or Resto Druid. You won't bring much to the table in 3s or RBGs though, as most people would just rather take a Rogue in those situations.
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