"[]" versus "[[]]" in the Bash shell

This can be answered, but I’ll ask anyway. I have two versions of script ( comp.sh) -

#!/bin/sh
export tDay=$(date '+%Y%m%d')
newfile="filename_$tDay"
filename="filename_20120821100002.csv"
echo $newfile $filename
if [ $filename = *$newfile* ]
then
  echo "Matched"
else
  echo "Not Matched!"
fi

Output:
$ ./comp.sh
filename_20120821 filename_20120821100002.csv
Not Matched!

and

#!/bin/sh
export tDay=$(date '+%Y%m%d')
newfile="filename_$tDay"
filename="filename_20120821100002.csv"
echo $newfile $filename
if [[ $filename = *$newfile* ]]
then
  echo "Matched"
else
  echo "Not Matched!"
fi

$ comp.sh
filename_20120821 filename_20120821100002.csv
Matched

Can someone explain to me why the difference?

Also - under what circumstances should be used [ ]against [[ ]]and vice versa?