Cameras vary widely in their convenience and capabilities, just as the people behind the lens also vary greatly. Wise people often say that you don't need a better camera in order to take good pictures. Maybe what you really need is some education in photography. Said in a different way, good pictures can be taken with virtually any camera. Even a pinhole in a box, as photography students are sometimes asked to show.
So does it really matter? Yes, despite that, it can make a difference.
First, the obvious. Some pictures are impossible to take with some cameras. I have a picture of a water ouzel. This is a bird that walks on the bottom of rushing mountain streams. I have a picture of one, but I doubt if I can convince you that the little brown dot in that stream is a bird. I can't really use that picture as anything but a joke. The picture was taken with a camera with a single fixed lens, around 50mm, probably.
I don't have a scan of that picture, which is in slide format, but below are two pictures of sloths. The picture at left was taken with the Mavica 91, which has a zoom capable of over 500mm on a SLR camera. The picture at right was taken with that camera with a 2x converter attached, which would be capable of around 800mm. Neither of these pictures would win a prize, but with a standard lens, you wouldn't even know it was a sloth, much less what a sloth face looks like.
So if you are a bird watcher or like to take pictures of other wildlife, a zoom lens is mandatory. Not only bird watchers like to take pictures of things that are far away. In my section called Slide film vs print film vs digital, I show several pictures taken from river ships. Without a zoom lens, your pictures will be almost devoid of interesting detail. Even normal cultural tourism benefits from a good zoom lens. Again, these pictures are only possible with a good zoom or telephoto lens.
So some things you cannot do with certain cameras. Other considerations may not be so absolute, but still need to be considered.