Also it's usually a good idea to add double quotes around the values; you will get very confusing error messages if either of the variables would turn out to contain an empty string (or a vast array of other possible problematic values).
if [ "$VAR1" = "$VAR2" ]
In real-world scripts, you frequently see a leading X added to the values, to guard against some of the remaining problematic scenarios (if $VAR1 has a value containing a dash as the first character, for example).
if [ X"$VAR1" = X"$VAR2" ]
For these reasons, I personally tend to recommend case over if for string comparisons.
To retrieve the week number of the file you can do something like:
VAR1='date +%V'
# get the week number of the file:
file_date=`stat -c "%y" file | cut -d " " -f1`
VAR2=`date +%V -d $file_date`
if [ "$VAR1" = "$VAR2" ]; then
....
else
....
fi
You should still use backticks for VAR1, too (ASCII 96, not regular apostrophes).
I was under the impression that the OP really did want to compare against the contents of the file, not the date on which the file was actually changed.