Well, here's a basic breakdown:
Articuno - Ice/Flying
Zapdos - Electric/Flying
Moltrest - Fire/Flying
Now right off the bat, Zapdos has an adventage over Articuno and Moltres as their Flying-types are weak to Electric. However Articuno gains an adventage over Zapdos because Zapdos' Flying-type is weak to Ice. And to finish this off, Moltres has an edge over Articuno because Articuno's Ice-type is weak to Fire.
Articuno: 1
Zapdos: 2
Moltres: 1
Now moving on to Resistance, Zapdos takes it as being part Electric-type makes it resistance to Flying, which all the Bird Trio is apart of.
Articuno: 1
Zapdos: 3
Moltres: 1
However this means nothing without good stats, so lets take a look at each ones stats:
Articuno:
HP: 90 Attack: 85 Defence: 100 SpAtk: 95 SpDef: 125 Speed: 85 Total: 580
Zapdos:
HP: 90 Attack: 90 Defence: 85 SpAtk: 125 SpDef: 90 Speed: 100 Total: 580
Moltres:
HP: 90 Attack: 100 Defence: 90 SpAtk: 125 SpDef: 85 Speed: 90 Total: 580
All are tied in HP and Total, but when it comes to the other stats most of them are different. From what I can see, Articuno is a defender, Zapdos is inbetween, and Moltres is an attacker. I'll give all of them a point for each stat they had then the other two (if there is a tie between two then both will get a point. If there is a tie between all three then none of them will get a point). So it looks like this now:
Articuno: 3
Zapdos: 5
Moltres: 3
To finish this off, I'm going to end it with what moves they can learn. I won't go through all the move but rather make my decision on two things: How many attacks does it have of its types & how many attacks can it learn that will hurt the other birds.
Articuno:
STAB: Looks pretty good for Ice, with it learning Ice Beam, Blizzard, and Sheer Cold naturally. But it's Flying attacks are about non-existant except for ones it share with the other birds (which means they doesn't count), so it will not be getting a point for that (but will get a point for it's Ice attacks)
Againt Other Birds: It knows a Rock move (Ancient Power) which is good, but so do all the other birds so it doesn't count it. Other then that, there is nothing Articuno can really use against its fellow trio members.
Zapdos:
STAB: It can learn Discharge and Thunder naturally along with Thunderbolt with a TM. Not onlt that, but it also knows a nice Flying move, Drill Peck, which means it will be getting both points for its types.
Againt Other Birds: If it wasn't bad enough that Articuno was weak against Zapdos' Electric-type, it's about to get worse for it. In Platinum, Zapdos can learn Heat Wave so it gets a point right there for that! Nothing more for Moltres though.
Moltres:
STAB: Flamethrower, Heat Wave, Fire Blast, Overheat, do I need to say anymore? And for Flying, Air Slash and Sky Attack just got it full points.
Againt Other Birds: But this is where it all ends, as it has no unique moves that it can really use against the other birds.
So with this in mind, here is the final total:
Articuno: 4
Zapdos: 8
Moltres: 5
So there you have it, looks like Zapdos takes the cake, with Moltres having a slice of it and poor Articuno being left out.