Smoothing is an operation that you can perform on your mesh if the edges are too rough and make your object look too low poly. In this example you can see that i created a cylinder and scaled it on the Y axis and then added some edge loops and extruded every other one out. This was to try and get a handle type shape for something like a sword or some other form of hand held weapon, However you can see the edges look far too rough and uncomfortable to be something someone would actual hold. So in order to correct this if you select the object and press 3 on your keyboard it witll smooth out your object in to a much more organic form.
Subdivision allows for you to add more geometry on to a given mesh. If you select the object and then go to edit mesh and then add divisions it will add more geometry you can also control the level of detail to a level of your liking. Subdivision is helpful as if you apply materials with displacement and don't have sufficient geometry it won't displace properly so when you subdivide it will actually add small geometric displacement. Subdivision is also great for a subdivision modelling work flow, this the process of blocking out the primary shapes of your object and then subdividing to add the smaller details to your model.
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