whatis displays one-line manual page descriptions, intended to give you a general idea of what a program do,
apropos searches the manual page names and descriptions, intended to help you to learn what program to use when performing a certain job.
Say, I want to know what does
df program do.
whatis gives me answer.
$ whatis df
df (1) - report file system disk space usage
And now, I want to unzip an archive.
apropos gives me some choises for consideration.
$ apropos unzip
bunzip2 (1) - a block-sorting file compressor, v1.0.6
funzip (1) - filter for extracting from a ZIP archive in a pipe
gunzip (1) - compress or expand files
lz (1) - gunzips and shows a listing of a gzip'd tar'd archive
preunzip (1) - prefix delta compressor for Aspell
unzip (1) - list, test and extract compressed files in a ZIP archive
unzipsfx (1) - self-extracting stub for prepending to ZIP archives
uz (1) - gunzips and extracts a gzip'd tar'd archive
apropos search man database, you can only get info on what's already installed on your system.