Ease of manufacture and repair are big selling points for rivets.
Imagine using a spot welding mechanism to build the aircraft structure (including the skinning) in the cover image of the video above. The spot welding electrodes would have to be able to reach approximately the same locations as the rivets, while moving around and through other bits and pieces of the internal structure, and possibly around and through equipment depending on what stage of the initial assembly process you are at. The initial assembly process might be able to incorporate such a method, but imagine trying to design a machine that could do such a thing on an aircraft the size and complexity of a 747. In addition, imagine trying to do the same thing - possibly in the field or at a smaller airport, or perhaps in the field at squadron level for military aircraft - when trying to repair damage.
Also, when a properly designed riveted structure begins to fail it is usually significantly easier to detect, using less sophisticated equipment, and subsequently more reparable than when a spot-welded structure fails. This is mostly due to the modes of failure for the different fastening methods - which unfortunately is a subject too complex (I think) to cover in detail here.
If you are interested, do a search for 'rivet failure mode in sheet metal' and/or 'spot-weld failure in sheet metal' (or similar word strings) for a fair number of papers and articles written about, and with descriptions of, the types of failure.