I work in the field on the theory/simulation side, and have actually had dinner and discussed research with Dr. Israelachvili a couple of times. I've only had a chance to skim the paper, but I think I can summarize it pretty well... by the time I've really absorbed it you folks will have moved on to the next shiny new story so I'd better do it now!
First of all, the report claims that the paper is all about how oil and water don't mix and makes a big deal about how we don't know how that works. For simple stuff like say water and a basic hydrocarbon like octane, that's really not true... it's all about what has already been said above, polar vs. nonpolar (electrostatics) and entropy.
Things get more complicated when you want to model something like an extended hydrophobic surface, or the interactions and formation of bilayer membranes like we have in a cell. It's been known from experiments since Dr. Israelachvili's work in the 80's that if you take two such surfaces (usually mica functionalized to make it hydrophobic) and bring them together in water, they will repel each other, up until at some point they very quickly strongly attract, expel the water between them and glue themselves together (also called "cavitation"). This is the sort of data shown in Fig. 2 in the paper. The connection with membrane formation is to describe how two membranes behave when they come close together, they have to do something similar to get close enough to fuse (figure 3).
Figuring out how to describe this behaviour from a theoretical standpoint has been very difficult! We know what all the parts have to be (hydrophobic,electrostatic, steric/Van der Waals, entropic) but haven't been able to put them together in the right way to describe all of the experimental data. What Jacob and his team have done here is found a nice way to 1) describe the hydrophobic interaction between extended surfaces mathematically (the equation above), 2) combine it with all the other parts (figure 4), and 3) show that the equation with a combination of fitted and measured parameters can fit the experimental data pretty well (Table 1). It's very nice work, definitely a step forward in our knowledge of hydrophobic surface and membrane interactions, and I'm going to make sure I study it more carefully soon!