When you are looking at your parts or talking about them with others the terms tend to get used interchangeably but they mean different things. When you cut a part you end up with a bur with very few exceptions. The process of removing that bur would be deburing obviously. That leaves you with a part that has no bur but can still have a crisp 90 degree corner. At that point you may be done or you could want to break that 90 degree corner and move into edge rounding. These things are not the same and are two separate tasks. While they can be combined they are two tasks none the less. Removing the bur and breaking the edges is about handling, welding, coating adhesion amidst other things and not about the visual appearance of the part.
Graining is much the opposite of deburing. In graining it is about the visuals more than any other. To get a good straight grain you need speed. You want the part to get from the front to the rear of a machine as quickly as you can to avoid the zig zag or wave pattern that can appear in the finish. The wave comes from the tracking of the sanding belts and while it can be minimized it can not be removed. In this process the edges at best will stay a crisp 90 degree but there should be no vertical bur. A side effect of the process is your leading edge tends to develop a "knife edge" where it can be very sharp. You have to take care when handling parts that have been grained as the risk of getting cut is there.
Finishing is a more general term that just references a final visual or physical result. Your finishing could be to get a #4 grain finish or a good random pattern for paint adhesion. It could also be that finishing for a part could be weld removal or even mirror polishing. It is a basic general term that references these final steps. This is where you can set the first impression for the person that receives the final part.
When you are looking at a process for improvements or even new machine purchases these terms become more important. Knowing what you need as an end result is a good place to start. If you need a rounded edge and a part ready to paint then you would not want to look at a graining machine as you will end up disappointed. Knowing the terms and the difference can put you in a better position in these situations and help to make better decisions.