I believe it does add damage. Sharpness acts as a damage multiplier. There's a formula floating around. Haven't got the faintest as to how big or how small a multiplier sharpness actually is (we talking 0.5 to 2, or more 0.95 to 1.05?). Would be interested in finding out actually.
They changed it a bit in this game so I can't give a definitive answer. Although as far as I know the only difference from Tri is they nerfed purple sharpness a little.
Here's a lazily copy/pasted chart from Grahf's old damage FAQ:
| Color | Raw [SHARP] | Elemental [ESHARP] |
|-----------+----------------------+---------------------------|
| Red | 0.50 | 0.25 |
|-----------+----------------------+---------------------------|
| Orange | 0.75 | 0.50 |
|-----------+----------------------+---------------------------|
| Yellow | 1.0 | 0.75 |
|-----------+----------------------+---------------------------|
| Green | 1.05 | 1.0 |
|-----------+----------------------+---------------------------|
| Blue | 1.20 | 1.0625 |
|-----------+----------------------+---------------------------|
| White | 1.32 | 1.125 |
|-----------+----------------------+---------------------------|
| Purple | 1.50 | 1.20 |
note: I think purple is like 1.40 raw now, or something like that. Still good, just not godly ridiculous.