The TUI B737 was overweight for the performance figures used, meeting the requirements at its actual weight could involve different speeds and power settings.
An aircraft is required to be operated within its certification limits, go outside those and you become a test pilot. Center of gravity comes into play as well if you are overloaded, particularly in the event of an engine failure. If an accident occurs, the loadsheet is one of the first things to be checked and the insurance company will be looking for an excuse to get out of paying.
A while back, I was a pax on an ATR42 operating out of a limiting airport and everyone had to stand on the scales during check in. The take off involved stopping on the runway and powering up against the brakes so I was quite happy to know that the pilot was working with actual weights.