Gaussian process smoothers (bs = "gp") in GAMs

In his inspiring overview of generalized additive modeling, Noam Ross makes a passing mention of gaussian process smooths (bs = "gp" in mgcv syntax), noting that they are especially well-suited for modeling variation in time. The description of the gp smooths in the mgcv documentation is rather terse and abstract, and I have yet to find a good discussion of exactly how a gp smooth differs from, say, a thin-plate spline, and why and how the gp smooth should be used for modeling temporal processes in GAMs. Can we have that discussion here?

Update
Gavin Simpson discusses gaussian process smooths and their application to time series modeling at length in this paper.