Why does my tip droop drastically when I turn my head to one side but not the other? (Photo)

It appears from your photos that your nose is deviated - this creates two different profiles due to a perceived "longer" side and a "shorter" side.  You will require a revision rhinoplasty to achieve a more symmetrical nose, and thereby eliminate your issue.  If your surgeon does not recognize the issue, or is unwilling to help, I recommend consulting with a Board-Certified plastic surgeon experienced in revision rhinoplasty.

Most people have a "good side" and a "bad side." This is from facial asymmetry. This goes for the nose as well. The reason why your nose seems to droop from one side is the photo is taken from the opposite side of the crooked tip. When you turn with the tip towards the camera, it foreshortens your nose and makes the tip look less droopy and the tip smaller. Rhinoplasty can help straighten the nose and make the tip appear smaller and less droopy from all angles. This is not exclusive to you, most people have asymmetric faces, even top models.