From Distillers Wiki
A continuous still doesn't have a large boiler which is filled with the wash. Instead, it is fed with a constant stream of wash through an injection point on the side of the packed column.
This essentially separates the column into two separate columns, the "stripping column" below the injection point, and the "rectifying column" above.
As the wash flows down the stripping column, the hot steam rising from the boiler strips the ethanol from the wash. When the wash reaches the boiler, it should contain little or no ethanol. From there, it flows out through an overflow and away.
Meanwhile, the ethanol and steam are continuing up the column. From here upwards, operation is identical to an ordinary reflux still.
Without a large boiler, warmup time is negligible on a continuous still, and heat losses from the boiler are reduced. Waste is disposed of as it is produced, rather than leaving you with a large tank of hot, spent wash.
A continuous still can process HUGE batches and run for a very long time non-stop. This is why a continuous still is the best for fuel production and why they are used by large distilleries.
There is one main problem with continuous stills, and it's the reason people choose the ordinary batch-run reflux still. This problem is commonly known as the "constant heads problem". The wash is not part of the equilibrium, rather it is fed in slowly. This means that you do not get the heads and foreshots at the start of the run as you would with a batch still, and they cannot be discarded. Instead, they are evenly distributed throughout the product.
This can be rectified by running the continuous product in a batch still and discarding the heads as normal. This run will be fast, as the ethanol is already highly pure, ~95%. This is why continuous stills are used for stripping.
Large distilleries which use continuous stills often have true fractionating columns as used in petroleum refineries, which can remove heads at a different height on the column during the run. Occasionally people undertake the effort to make a continuous column at home that can remove heads and methanol during the run, but as of now there are no documented working designs.