Weaknesses
Mechanical connections such as nut and bolt applications and heavy machine screws can be incredibly strong connections. However, they are intrinsically weaker than continuous metal. Compare it to a welded joint. Molten steel, for example, coalesces into one continuous piece. There are no threads to strip. Vibration cannot loosen it, and stress is less likely to break a properly executed weld than a bolt.
Difficulty of Refastening
A benefit of mechanical fasteners is the ability to be fastened, then refastened. This can be a drawback, too -- particularly for some kinds of mechanical fasteners. Engines, for example, are full of bolts that are time consuming to remove and retorque. Although, other mechanical fasteners, such as the buckles found on ski boots, can expedite fastening and refastening.
Forced Precision
Most mechanical fasteners are very precise, usually consisting of a machine-threaded cylinder and machine-threaded bolt-shaft. This is a drawback when you need a more "approximate" connection. For example, many motorcycle tank bags use powerful magnets. The magnets let you set the tank bag on the tank and it holds it firmly, wherever you place it, but not so firmly you can't pull it off with a little tug.
Corrosion
The temporary mating of metal surfaces -- in spite of a precision fit -- is still vulnerable to corrosion, most often iron oxide, commonly known as rust. There's little use in having a temporary mechanical connection if it becomes permanently rusted together. If it happens, you lose the benefits of the mechanical connection, without the advantages of a welded or other kind of connection.