Some reasons you might choose PAPER:
You want to feel less precious about what you are making.
You want the end product to feel like less of a solid object.
You want to paint in thin, watery layers.
You need that bumpy texture of "rough" surface paper.
You want a cheap surface to do some fast and expressive pieces that are a bit experimental - You'll probably throw some of them out.
You want an incredibly smooth surface for fine lines - Hot Pressed Paper.
Unless the paper is 300lb thickness you may not want to work on large sheets because thinner paper damages so easily when you are trying to store it or move it. Solution - frame anything right away and/or be meticulous about how you move and handle it
* Should be framed under glass.
Some reasons you might choose CANVAS:
You want to go big and canvas is durable and unlike panels - won't weigh a tonne at a large scale. Unlike paper it is ready to hang in a pinch - no frame required but always recommended when you want to put your best foot forward and impress people with your seemless presentation.
Can be framed in a front-loading or back-loading frame.
You have the option of taking it off the stretcher bars and rolling for shipping which can save you a bundle. Test first - some paint surfaces will not do well when rolled - depending on application and mediums used rolling can cause cracks and delamination.
Some reasons you might choose PANEL:
You want a rigid support so you can do aggressive application with a palette knife, this is too risky with canvas as you can end up poking a hole in the canvas if you are too aggressive with your palette knife.
I find it gets a bit too heavy for me when I start working on a larger scale panel so I reserve large scale for canvas.
You want to use certain mediums and applications that require a non-flexible support. Some of those techniques and mediums will easily crack if you try them on canvas which has a certain amount of flex to it.
You want to expose the wood grain in your image.
You are creating an image with fine-lines and prefer the lack of bounce/flex to doing this type of work on canvas. It can offer more control in this regard.
Like, canvas you can hang it up on the wall "sans" frame, ready to sell.