Wikipedia states that a lumped parameter model "simplifies the description of the behaviour of spatially distributed physical systems into a topology consisting of discrete elements". A multi-compartment model seems to be exactly the same thing, calling those discrete elements "compartments", but Wikipedia states that a multi-compartment model is a lumped parameters model.
What is the (I assume subtle) difference between the two? Are there lumped parameters models which are not multi-compartment models?
Both is very close I would say. Lumped parameter model is mathematical model based on ordinary differential equations, where space variable dependance is not present (if you consider a PDE model and further no dependence on space variable, then you get a lumped model). If the modeled system is made of compartments (once I saw a numerical model of human body, where each organ was a box described by one ore more ODEs), then you have a compartment model, which is lumped at the same time. On the other hand if you consider for example an elastic body described by the law of elasticity including temperature dependence (with some boundary conditions, with unknown displacement $u_x$ and temperature $T$) and you consider zero dependence on space variables (i.e., you consider a solid point), you get a set of two ODEs for displacement and temperature. This is definitively a lumped model and it can be theoretically considered to be a compartment model with one compartment only.
In conclusion I would say: both names describe a mathematical model based on ODEs, however the word compartment model has sense, if the modelled system is decomposed into more compartments, each of which described by one or more ODEs.