Instructions
Try to catch the rogue with a cat form Pounce while you are both stealthed. The Pounce stun gives you some time to set up your next moves and gain some distance if needed.
Apply the spell Faerie Fire or Faerie Fire Feral as quickly as possible, either once you find the rogue in stealth or once you recover from his initial stealth opener. Faerie Fire reduces the rogue's armor and prevents him from stealthing unless he uses Cloak of Shadows. If you are a "Balance" or "Restoration" druid, keep Insect Swarm on the rogue. Use Cyclone liberally to control the flow of the battle.
Adjust your offensive measures to your talent spec. "Restoration" druids will ideally have another player character around to support this spec, as it does not have the burst damage to easily kill a rogue one on one. Using Roots, Moonfire and Starfire along with travel form for kiting gives the "Restoration" druid a slow but easily survivable way to defeat the rogue.
"Feral" druids generally need to stay in cat form to control the rogue. Stack your DoT attacks on the rogue and keep Mangle up at all times. Build your combo points through Mangle so you can consistently land Rips.
"Balance" druids rely on Moonkin form and either a Solar or Lunar Eclipse state to deal massive damage to the rogue. "Balance" druids are particularly vulnerable to the rogue's spell-interrupting Kicks, which can be used every 10 seconds. Cloak of Shadows is another nasty ability the rogue uses to become invulnerable to magic damage. Keep the rogue at range as much as possible while building up to an Eclipse state.
Keep yourself in a survivable situation. Druids are a very resilient class, with many tools to survive a high and sustained DPS attacker such as the rogue. "Feral" druids gain additional health and armor through bear form, as well as Bash. The Bash stun gives you enough time to switch out of form and heal yourself without risking an interrupt from the rogue. "Restoration" and "Balance" druids can use Insect Swarm to debuff the rogue and Roots to keep their distance. Travel form is particularly helpful with caster druid kiting, although bear form does have its uses as well. Tree of Life for "Restoration" druids can also tip the scales, as the druid can heal through a great deal of damage quickly with this form.