Normally the term 'freeze distillation' loosely applies to freezing a liquid and then retaining either the solid or liquid as a product while disposing of the rest.
Traditional distillation uses the equilibrium distribution of chemicals between the liquid and vapour phases to separate them. Or: at
a given temperature and pressure, a saturated solution (ie at it's boiling point) will exist as both liquid and vapour. If there are two different chemicals (A and B) they will distribute themselves between the liquid and vapour phases based on their volatility (boiling points). If A has a lower boiling point then there will be more A than B in the liquid and less A than B in the vapour.
This preferable partitioning into either liquid or vapour is repeated as a counter-current flow in a distillation column; liquid flowing down getting enriched in A at each stage and vapour flowing up getting enriched in B at each stage.
The same could be applied to the equilibrium partitioning between solid and liquid phases. Ie. small solid particles brought in contact with liquid in a series of counter-current stages (similar to solid-liquid extraction). Where the more melty chemicals get enriched in the liquid phase and more freezy chemicals get enriched in the solid phase.
The advantages of this are:
1) Distillation of close boiling components require a large number of stages (big column) or reflux (big energy) or advanced distillation like reactive/extractive/azeotropic/vacuum (big pain). Freeze distillation could bypass this problem.
2) Components with much higher enthalpies of vaporisation than enthalpies of fusion (think water) would take less energy to bring to saturation.
3) Components with high boiling points will cause higher heat losses, may destroy certain chemicals, blah blah blah freeze distillation.
Obstacles (or how I learned to stop worrying and why this isn't used)
1) Solids handling is a problem, you can't pump it, you can't compress it. It wears away at machinery. When you freeze something it is thermodynamically favourable for it to form on the walls of the tank rather than dispersed in the liquid.
2) Solids don't diffuse good. In a liquid droplet, if it is small enough, you can assume it is well mixed. So when volatile components boil out from the surface the remainder is evenly distributed through the droplet, ready to boil again. In a solid, if it a component is not on the surface, it will be trapped.
3) Solids (usually) don't conduct good (I suppose I'm thinking of ice here). You will get temperature gradients and different components freeze at different temperatures (the whole principle behind freeze distillation) so you will get an impure solid product.
2 and 3 are reasons for small particles - something that would be very hard to control. (sorry it's so long but I typed it out twice [Mozilla fail] and I'm still struggling to explain myself clearly)
Oh and the principle of different chemicals partitioning preferentially into solid or liquid phases finds practical use in Zone Refining (see link) used to homogenise some (expensive) alloys.