When I tell people I run with my dogs they, most of the time, assume I do canicross. And when I talk about canicross, they usually ask me if that means I run with my dogs. Yes and No. In practice yes, technically no.
Canicross begins with running with your dog, indeed. But further it's a sport and it involves competition, has many other rules than just "a human and a dog running together" and also requires appropriate material.
Canicross is also about running short distances in a cross field - and not on the asphalt. Another important aspect of it is that the dog should run in front of the human and is supposed to pull him.
On the other hand, running with your dog has no specific rules. You might run wherever and however you want - as soon as you are not putting your dog's health in risk, of course!
It can be a (very nice) hobby but it can actually also be an important part of the canicross training. Long and easier runs (with less pulling, focus on the commands or on his ability to listen to you for instance) are also fundamental to increase performance in short and fast runs.
Finally, you might also have heard about canitrail. If you know what trail running is, you might also be able to guess what canitrailing is about. For those who never heard about it, trail running also means running through the fields and nature but unlike cross, the distances are longer, the paths are not always obvious and the time or speed are not the most important aspects; The pleasure and joy are! - And of course your dog must be in good shape in order to safely enjoy the trail with you. That's why simply running with him might be also crucial.