Big / Large

BIG / LARGE:
Notice in the sign below the "L" hand index fingers are bent?  That is a generic sign for "largeness" or the state of being big.  If I use "normal" "L" hands then the sign becomes an initialized sign for "LARGE." 

The larger you do the sign, the larger whatever you are talking about.

Note: if something is "very" large, you might also use the mouth morpheme "CHA."  You do this by mouthing / saying "cha" (on your lips not out loud) while doing the sign.

BIG:
The sign for "big" is made by forming the letter "b" with both of your hands in front of you, palms facing inward. Move both hands away from each other. Move your hands beyond the width of your body - the further the distance, the larger the word described. When you do this sign with a "B" handshape, you are using an initialized sign. Many people consider this "Signed English" rather than ASL. But notice, I modify the "B" shape so that the thumbs are loosely alongside the palms and I curve the handshapes just a little bit.

Large:
If you use normal (not modified or bent) "L" handshapes, then you are doing an "initialized" version.  This sign is considered to be "Signed English" rather than ASL.
Note: Palms face toward each other (not out).