What kind of problems can a flight have if passenger weight is miscalculated?
In general, an incorrect takeoff weight can have serious consequences for an airliner, especially when the actual TOW is close to the RTOW (regulated takeoff weight, see e.g. this question: What is RTOW and how is it different from MTOW?). It is possible that the aircraft is unable to takeoff within the available runway length. This is particularly important in case of an engine failure during the takeoff, since normal performance with both engines operating typically has enough margins. Even on a long runway, the pilots might rotate too early because of an incorrect rotation speed (VR), which could result in a tail strike.
In this particular case, the difference in performance was minimal. The AAIB (Air Accidents Investigation Branch of the UK) investigated the incident and concluded:

Whilst an incorrect takeoff weight was used for aircraft performance planning, the thrust
required for the actual TOW and environmental conditions (88.2% N1
) was marginally less
than the thrust used for the takeoff (88.3% N1
). This meant the safe operation of the aircraft
was not compromised.

The report also shows the difference in the performance calculations with the incorrect (left) and correct (right) TOW:

The difference in the V speeds for takeoff is just one knot, which is not significant enough to have any adverse impact on the takeoff.