Each morning I scan my computer with AdAware's scan system (Anniversary Edition) and it immediately shows that the last scan date is updated to this date.
However, when I turn off the computer and then turn on again, I get messages saying either "Scan Recommended" or giving a last scan date of some weeks ago.

What is the problem? Why does it not show the current date of scan?

This is a known problem, although it doesn't happen to everyone and not all the time. It happened to me once, then never again. Member ColTom tested 2 systems extensively and could not always replicate the problem. Lavasoft has determined the cause to be the AAWService not shutting down properly, therefore not writing the last scan data. They're probably not going to release a fix just yet since there's a new, major update due out later this month.