The Channel 4 TV documentary Power Into Art (and presumably the resulting book) explained that the Tate Modern architects wanted the bridge to land in a certain way in relation to the Bankside powerstation and this conflicted with Norman Foster's design.
NF wanted it to land at right-angles to the river and come up more-or-less to the brick wall and/or an entrance. However the actual north side entrance is upriver of that position and Herzog & de Meuron wanted the north and west elevations to stand clear of anything else.
Consequently NF had to redesign the southern landing, changing it from being essentially identical to the simple northern one.
The bridge walkway splits into two as it crosses the river bank and then turns back on itself with a ramp down between the split, leading down to the riverside path.
This somewhat awkward and contrived design must have an effect on stability. For a start the contact area with the ground is presumably much less.
In short, I'm not surprised it's had a stability problem, though it does seem far more severe that it should be.
The engineers so far are saying it's the fault of people walking in time to the swing - yet while it sways people have to. It's all rather circular! Clearly damping is needed as a first step.
The bridge reopened at 10am on Friday 22 February 2002.