That's a strange combination, but perhaps not as much as it might seem. Old-time music is often called "hillbilly trance music," and Eno's work is not cold and analytical the way a lot of electronic music is, and in fact much of it is not electronic at all, but organic sounds sampled, processed and reused.
Furthermore his approach to music is warmly human and wonderfully open and unconstrained -- a far cry from the rigid orthodoxy of many musicians in both old-time and electronica. The more I've read his writing, and read about him, the more I've grown to really like him and his approach. I was pleasantly surprised some months ago to find that he's a fan of simple singing, and as Tamm says, his approach to music is one of "sustaining an open mind and childlike curiousity about the infinite range of musical possibility."
While some deride his music as "still waters that don't necessarily run deep," or criticize his naivete, I actually find his experimental work much more listenable, much warmer, than most of the people some would consider his peers. Eno has a sense of humor. His electronica and ambient music, unlike that of many of his imitators, is emotional and organic, more varied, more sustaining of interest, than classical minimalists or experimentalists like Tangerine Dream. At the same time it's more thoughtful and less obvious than even the sophisticated electronica/pop of, say, Massive Attack or St. Germain. As Tamm says, Eno has maintained a "sense of wonder," unlike so many others who work too hard to appear urbane and sophisticated and detached.
I've had an interesting relationship with his work over the years. I much prefer his songs to his instrumental and ambient work, while he largely seems to have lost interest in songwriting, in general preferring to wander away from the melody and lyrics that occupy the foreground and focus your attention on the horizontal motion, and instead wander into the background to experiment with textures and colors. "The problem is that people, especially people who write, assume the meaning of the song is vested in the lyrics," he says. But for him, "music in itself carries a whole set of messages which are very, very rich and complex, and he words either serve to exclude certain ones of those, or point up certain others that aren't really in there, or aren't worth saying." 
Over the last couple of years I have found myself returning to his ambient work, and rediscovering it. I think I became disenchanted, to some extent, having followed what I thought was a progression from his work to Tangerine Dream and the Yellow Magic Orchestra and various ECM jazz artists and Kitaro and Philip Glass's more angular minimalist work. After the initial fascination, that music wore think very quickly. Later in college, I got into blues and reggae and hip-hop and forgot all about the other stuff. A lot of it ended up in used-record stores, and none of it is among the music I wish I hadn't sold.
But I never sold any of my Eno albums, and his work, along with that of the musicians he works with frequently -- Robert Fripp, Jon Hassell, Harold Budd, Daniel Lanois -- have never really left me. It's no more related to ECM jazz and space/new-age music than Coltrane's is to Kenny G's.
I've done a lot of flying in the last month, and I have been listening closely to two of his ambient albums, Music For Airports and The Plateaux Of Mirrors (a collaboration with pianist Harold Budd), sitting on a planes or in airports and doing nothing else. They support this listening. They're not boring; they're almost fractal. The more closely you listen the more you hear. Eno says at one point, talking about the simplicity of his music, "It's not because it's simple, any idiot can do it. There's sensitivity in the way you can strike just one note." 
And yes, you can pay attention to ambient music. As Eno says, he wasn't trying to create wallpaper or background music, but music that could be listened to in the background, that did not demand your attention or try to push you in a particular direction. It's ambient in the sense of trying to create a space, an ambience, and allowing you to find your own place in it.
The first track on his first ambient album, "1/1" on Music for Airports, is an amazingly beautiful piece of music. It's nothing more than a repeated simple piano figure, varied slightly, with pauses in between repetitions. But it's never played the same way twice; there is tremendous feeling in the dynamics and the variations and the timing. Eno processes it all heavily, but not in a synthetic or overbearing manner; at first you might think you are listening just to a piano. He processes it gently, playing with the attack and decay, adding nearly inaudible washes of keyboard, or long sustained bass notes, sometimes harmonizing with or repeating a note with the synthesizer.
You hear every single note, and Eno's treatments and effects result in endless echoes, different every time, different colorations, subtle harmonies. "I often sit at the piano for an hour or two, and just go "bung!" and listen to the note dying," he says.  "Each piano does it in a different way." That's not electronica, that's the organic and unpredictable behavior of wood and wire. "You find all these exotic harmonies drifting in and drifting out again, and one that will appear and disappear many times. There'll be fast-moving and slow-moving ones. That's spell-binding for me."
His treatments amplify that focus, and with echo, delay, and nearly inaudible small noises, he places the entire piece in a very physical space. I want to say it gives a feeling of clean and perhaps stark spaciousness but perhaps that's just suggested by the title. But it is very evocative music, warm and physical and compelling.
What he's done on this piece is almost a meditation on the tones and possibilities of this simple figure. It's repetitive, but not repetitious; like many fractal structures (tree leaves, coastlines), it looks the same from a distance, but no matter how closely you look there is always another level of detail to see.
"The thing permits you any level of scrutiny," he says of the structures he prefers. "Things that allow you to enter into them as far as you could imagine going, yet don't suddenly reveal themselves to be composed of paper-thin, synthetic materials." 
Today while my flight was landing in Seattle I was breaking the law and listening to a recording of Robert Fripp's performance in the Winter Garden less than a year before it came close to being completely destroyed on September 11, 2001. It was a lunchtime performance, and I walked over from my office two buildings away, and sat in a front-row seat. He sat, as usual, black-clad and alone, silent and nearly motionless, on a performance stool.
He held his black electric guitar in the classical fashion, with a rack of effects units next to him and a pedalboard at his feet. He struck a bell-like note and held it, captured it and successive notes in a loop so they played continuously, building up layers and layers of sound that filled that enormous space. He would play nothing for moments, then splash chiming clusters of notes over this palette. He played organ-like swells or deep bass tones, casting long arcs of music out among the palm trees and the spectators that bounced off the glass and the marble and came back to him. And he answered that return, listening and responding and building a structure that was almost visible. It is not melodic music, but spatial; its movement is physical rather than harmonic. Even though it is created with digital equipment and an electric guitar, it is deeply organic, rooted in the place it was created, one musician's response to that day, that place, that audience.
With his professorial look, his silence, and his inward focus, it's very easy to mistake him for a technician. But he's not; this was a beautiful spontaneous act of creation that touched everyone. It was a grey November afternoon, the Hudson choppy and cold beyond the empty waterfront walkway. Premature holiday decorations adorned the shops around the Winter Garden. Tourists and traders stopped to listen, standing on the balcony or walking past with their lunches in bags from Donald Sacks. Some quickly lost interest and walked away, but more than a few people were completely captured by the flowing music, as warm and beautiful and evocative as any chamber music performance I'd ever seen in that space. Maybe more so, because rather than struggling against a too-large space not meant for music, Fripp was adapting to it, fitting it, using it as the ultimate analog delay unit, making his music part of the space and the day. That was ambience. The recording (legally available for purchase from Fripp's web site) does not entirely capture that moment, but it is beautiful and spacious and I am certain it is not just my memories that make me hear the glass and the marble.
Meanwhile, I'm heading back to West Virginia so this is the last you'll hear about electronic music for a while.