The question comes up, from time to time, among those who practice both arts, when would you use one as opposed to the other? My answer (for the time being, as always) is that it depends on my relationship to uke's "sphere of influence."
First of all, let me define what I mean by "sphere of influence." I imagine it as a giant bubble that surrounds each of us, and it represents a section of air or space where my hands (primarily, although it includes feet, too) have the greatest ability to affect something or someone else. The outer edge of that sphere, then, would be the furthest extent of my reach. If I hold my hand out at arm's length and I can touch it, it's on the edge of my "sphere" (also, if I can stretch out my leg and touch it with my foot, it's on the edge of my sphere).
Now, the thing to remember about this extreme outer edge is that, while I can touch something or someone, I can't really do a whole lot with it. I certainly can't deliver a whole lot of power. You don't open a jar of pickles while holding the jar out at arms length, right?
To start deliver a significant amount of power, my target actually has to lie a few inches within my sphere. For example, try to kick a punching bag that's at the very outer edge (or punch it) where you can just barely touch it. You scarcely budge it. Scoot an inch or two closer, and suddenly you can kick/punch the crap out of that bag.
But here's the interesting thing: move even closer. Suddenly, your fist and foot can't deliver the same power. The bag "stuffs" you and you eat the energy and stumble backwards, even. This realization is why muay thai fighters begin to employ their knees and elbows. Shorter "weapon", smaller sphere.
At this point, uke can't really deliver as much power. About the best he can do is start to curl his arms back towards himself.
Aikido, for the most part, tries to stay at the outer edge of this sphere, where uke's power or influence is pretty weak. Notice how uke's sphere of influence fades as we get behind him. Imagine as well that it fades whenever we get on the outside of either arm; aikido thrives there, too.
On occasion, however, the aikidoka will notice that, for whatever reason, he's wandered into uke's sphere. He's in very real danger here. Much of the time, then, his best option is to keep moving inward, slipping inside uke's sphere, where he's weaker again. This gives us the irimi or "entering" ideas: irimi nage/aiki nage, tenchi nage ("heaven and earth"), forms of gyakugamae ate where you're inside uke's arm as in san kata (instead of outside as in the 17), even sumi otoshi.
But aikido's repertoire in this condition is relatively limited. Which, to me, is where much of judo comes in. I somehow end up in front of uke (not behind his arm as I'd like), and I've wandered into his sphere of influence. Best to keep moving in, join with his center, and find a throw.
When the whole Ultimate Fighting Championship business started out, we saw a lot of judo and mostly BJJ guys destroy all the punching kicking guys, right? They just started at the edge of the sphere, moved inward, weathering the storm of kicks and punches until they made it inside the sphere, and then proceeded to knock their opponent down and own them on the ground (where they had no training whatsoever). All of which, naturally, gave birth to the idea of "mixed martial arts". Learn what to do with a guy when he's in your sphere (kicking, punching), but also when he's inside (standing and on the ground).
Once inside, as I mentioned, we can't really deliver much energy outward (punching, kicking), standing or on the ground. And since our arms start curling back toward ourselves in order to do anything to the guy, we get into concepts of controlling uke's movements, primarily through osae komi, or hold downs (choking and arm bars extending out of the idea of first establishing some control).
At least, that's my understanding at the moment, anyway. So far, it's given me a nice frame of reference where I can at least start to view the arts in the same frame, as opposed to two separate, almost competing, entities.