Are biking and running the same thing?

Well, to get things straight, I know that biking involves a bicycle, so...

From a hypothetical point of view, does bicycling for one mile have the same healthy effects as running for one mile, or does one differ?

Plus, does bicycling work on different parts of the body, or does it work on the same parts of the body that running works on?

It is different. I wouldn't compare the two in distance because it is much easier to ride a bike for a mile than to run it, time wise they are much closer although I'd say it takes slightly longer to get the same workout out of a bike. And yes it does work different parts of the body or different parts of the same muscles. Advantages to biking are one, you can go longer distances which can keep things more interesting and two, it's much easier on the knees. a good third of the bikes i sell at work are to old runners who want to stay fit after they develop knee problems.

mile per mile running will burn more calories. But biking is non impact so it will not wear down your joints like running will. To me it feels like i use more of my body when biking. Especially when you are hill climbing, you are using your upper body and lower body to push and pull your way up

All of the above.

Cycling is certainly kinder on the bod, but you have to go farther to get the same benefits....muscle wise. Cardio wise, it "may" be about the same....it all depends on the terrain you're riding in.

I found that I could cycle many, many miles..... but walking at olympic pace really gave my leg muscles a workout....not my cardio cause I could go forever.... but my muscles... oh boy!