The differential and total cross sections for kaon pair production in the pp->ppK+K- reaction have been measured at three beam energies of 2.65, 2.70, and 2.83 GeV using the ANKE magnetic spectrometer at the COSY-Juelich accelerator. These near-threshold data are separated into pairs arising from the decay of the phi-meson and the remainder. For the non-phi selection, the ratio of the differential cross sections in terms of the K-p and K+p invariant masses is strongly peaked towards low masses. This effect can be described quantitatively by using a simple ansatz for the K-p final state interaction, where it is seen that the data are sensitive to the magnitude of an effective K-p scattering length. When allowance is made for a small number of phi events where the K- rescatters from the proton, the phi region is equally well described at all three energies. A very similar phenomenon is discovered in the ratio of the cross sections as functions of the K-pp and K+pp invariant masses and the identical final state interaction model is also very successful here. The world data on the energy dependence of the non-phi total cross section is also reproduced, except possibly for the results closest to threshold.