This is really cute and wonderfully drawn. The colors are bright, strong and work together perfectly. I also like the shading technique you use, it works extremely well with your style and this sort of picture.
The anatomy of the birds is also correct, so there isn't much I can say to that, but one thing that would have been nice is if their beaks weren't the same color as their feathers, just to make it stand out a little more.
I would actually also suggest adding just a hint of a background, instead of leaving it purely white. Maybe a very gentle touch of red to help the birds pop out more, but that's a personal opinion, what matters is what you like more.
You've done a wonderful job on both of them. They both have quite a bit of character to them. However, after looking at them for a while I think the copic markers have a better range of colours in this particular instance. The comparison of them side by side and facing each other make for an effective piece of art. Because they are facing towards each other the viewer is also drawn to look at the birds instead of off of the paper. The spot of white on the copic bird defines the eye a bit more and allows it to have so much more life to it then the watercolour bird. The different tones from the copic marker make for a much more interesting subject to look at. If you could give the watercolour bird some more variation in colour in general, it'll greatly improve it's effectiveness in the picture.
The finer wing feathers on the copic markers are also quite effective. They make for a more realistic picture as they blend in more with the animal while still giving the feathers that extra definition needed.
The pen markings have been wonderfully integrated into the birds, they clearly show the shape of the bird while not intruding on the colours. They also help to give them some character and perhaps even a cartoon style to them while still retaining their realistic feel.
I quite like the ground you've put in as it makes it more of a complete picture rather then just a comparison. The lack of background works well in this case, as it doesn't change the look of the colours of the birds.
The best thing about this picture is that it combines two very different media and makes them hardly distinguishable. Each medium has its own set of properties and characteristics. It's hard to bring out the best of each of them, but it's even harder to pair them up to portray the same style. Without a detailed analysis, the differences between the watercolour and the copic is hardly noticeable, and if it wasn't for reading your title and description, I would've never known. Being able to match up two very different media is very praiseworthy, and because you did it so professionally, I tip my hat off to your skill and technique.
The other really nice touch are the white highlights. It resonates with the sketch-style black outline well, and it gives the two birds quite a bit of contrast. Good job!
But even so, there are some details that reveals the difference in the two birds. For example, the copic marker bird is a lot more detailed, especially in terms of colour layering, than the watercolour. By this I mean, the copic marker bird shows traces of green, yellow and orange in its blue feathers, and the blue used ranges from navy to cyan. The watercolour bird, on the other hand, is predominantly navy, and besides the flank, the blue only varies in saturation rather than hue. This, I understand, is probably the direct result of watercolour being much harder to control on a small area than copic marker, which makes it harder, if not impossible, to add so many layers of colour to the bird. I have never used watercolour, so I don't know its limits, but surely you do. So while knowing that, perhaps it is worth considering adding a few more variations in colour on that front.
Last thing is the birds' eyes. The copic marker bird's eye is highlighted with a glare, which makes the bird rather lively, but this effect is not mirrored in the watercolour bird's eye. The lack of the glare makes that bird look a bit stiff, its eyes rather hollow. I know of a few ways to add this effect in watercolour, and since you're far better at watercolour than I'll ever be, I'm sure you know many more. Maybe with masking fluid? Maybe simply with the white pen that you used to draw the highlights? Whatever the method, I'm sure adding a bit of highlight in that bird's eye will suddenly bring it to life.
All in all, this is a very masterly piece. I am thoroughly impressed with your ability to wield both watercolour and copic markers with such competency. Great job!