Multi-scale modelling refers to the modelling of a physical problems which involves the description of different time or space-scales. Think for example to the modelling of nonequilibrium dynamics of electrons and ions initiated by ultra-short laser pulses in condensed-matter.
At first a lase pulse of few femto-seconds (fs) may excite electrons in a nonequilibrium configuration. There will be a coherent electron dynamics on the time scale of few hundreds of fs. Coherences will eventually de-phase leading to non-coherent and non-thermal electron population dynamics. Electrons will reach a quasi-equilibrium statistical distribution at very high temperature and, on an even longer time-scale cool down. Due to the interaction with the lattice, also the phonons will evolve, in general on a longer time-scale compared to the electrons. There could be coherent phonons at first and later non-termal phonon distributions, phonon heating and cooling. On even longer time scales both electron diffusion may take place, as well as structural distortion of the material. See here for more info: Nature Materials
While a fully quantum mechanical approach could describe all these phenomena, the approach would be non feasible for all the phenomena (and also not needed). The fully quantum description would be useful to capture the initial coherent dynamics, Boltzmann rate equations to capture the non thermal occupations, effective temperature models to describe the cooling processes, etc ...
However each of the above mentioned approaches works on a specific aspect and on a different time scale. This means that there is a time (and space) scale mismatch. Putting together different kind of simulations is what I would call a "many scales" approach, i.e. you kind of partition the physical process in sub-problems. What a real "multi scales" approach would do instead, is to somehow interface the different approaches and enable one method to continue from the output of the previous. A "multi scales" approach would enable "scale bridging" physical problems with a "scale mismatch".