Let's compare the two:
Robes: Tie. Both their robes are equally useless in the fight.
Gandalf gets points for style, though.
Staff/Wand: Tie. I'm going to assume that the Elder Wand's supposed "cannot be defeated in battle" powers do not work on Maiar.
I think that's more of an urban legend, but it's a powerful wand nevertheless so w/e.
Swords: Sword of Gryffindor, hands down if we're working on the assumption that this is post-Chamber of Secrets Dumbledore. That Basilisk venom infused into it is no joke. Otherwise, tie since there are no orks present for Glamdring to work on.
On the other hand, Gandalf is actually skilled with swords. Dumbledore's the squishy mage archetype up to 11.
Animal Companions: Tie. As per the stated conditions, it's just those two. Gandalf does not get to bring Shadowfax and Dumbledore does not get to bring Fawkes.
Oh, they're there. They're just busy having a sweet-ass [
s]Ride animation in the background.
Summoning to Giant Eagles: Dumbledore wins this one, oddly enough. He can transfigure them out of rocks while Gandalf has to send a butterfly to ask Gwaihir for help.
I thought
Deus Ex Machina was Gandalf's personal spell? :V
Ability to Make Light: Tie. Maybe slightly in favour of Gandalf, but I'm pretty sure Dumbledore can do Lumos wordlessly.
Gandalf has the disadvantage of wearing white, so if light would become an issue then he would be easier to spot.
Shoving People Around By Pointing Sticks At Them: Tie. Gandalf does this with his staff, Dumbledore does it with Depulso.
Alternatively, hollywood-shotgun at close range :V
Ability to Fill Swords With Lightning:Gandalf wins this one. No Harry Potter character ever does this.
Dumbledore probably can negate those +2d6 electric damage, though.
Shooting Lightning At People:Again, no Harry Potter character actually seems to shoot lightning at people.
But they're very good at casting
Magic Missiles.
Blinding People With Bright Lights:Tie. Gandalf can do this, and Lumos Solem is apparently taught to 11 year olds in the Harry Potter universe.
One would think that, being wizards, they would be prepared for that kind of stuff, but I just cannot imagine Dumbledore wearing sunglasses :V
Magic Shields That Block Stuff:Tie. Gandalf does this against the Balrog and Dumbledore definitely knows Protego.
Gandalf's is probably stronger though, what with being able to block a goddamn monstrosity made of shadow and fire.
Fire Summoning:Victory here to Dumbledore, who can do Fiendfyre, which cannot be put out with water or magic.
Gandalf started wearing a robe of fire immunity ever since Moria :V
Destroying Bridges While You Stand On Them: Tied. Both are equally capable of doing so, Gandalf with his staff and Dumbledore with Reducto.
Gandalf does it with more style, though. On the other hand, Dumbledore would be less at risk of falling down the bridge himself.
Teleporting: Dumbledore can Apparate. Gandalf cannot.
Wizards always arrive on time. If one chose that "on time" meant "within a second"... :V
Turning Invisible:Dumbledore can use the Disillusionment Charm. Gandalf cannot.
I'm pretty certain that Gandalf has
Glitterdust prepared for that exact reason.
Mind Reading: Dumbledore has Legilimency. Gandalf does not.
It's not something that you can use in the middle of a duel, though. Also, Gandalf not being human, it might not even be safe.
Transfiguring Bits of Dirt Into Tigers: Gandalf cannot do this; Dumbledore can.
Joke's on you, Gandalf has a few hobbits up his sleeve. Literally :V