I don't like how wide my face looks when I smile, especially my nose. Why does my nose expand when I smile? (photo)

The base of the nose is made of cartilage in skin and is therefore malleable. Facial expression including smiling and grimacing can distract this structure laterally. This is normal.

Hello and thank you for the question. 
The short answer to why your nose appears wider with smiling is that the geometry of your nasal tip cartilages change with smiling. Although the picture is limited to a single view I would guess your nasal tip droops a little with smiling if we had a lateral view. This is due to a small muscle (depressor septi nasi) connecting the nose to the upper lip at the midline. This can be treated with a very small amount of botox or during a rhinoplasty if you are seeking any other enhancements to your nose. 
Your upper lip can also be addressed to help with you smile. The lip is divided into the the skin portion and red portion, where you apply lipstick. Your upper lip has a thinner red lip portion and more prominent skin lip. You can try having an experienced practitioner precisely inject a small amount of  botox  in the upper lip to help with a subtle lift that will also enhance your smile and teeth show. However, in my opinion your best option is a lip lift -  a minor surgical procedure that reduces the skin portion of the upper lip. This will help you considerably. It will enhance the ratio of your red lip to skin lip and also show off your new beautiful smile. 

Hope this helps and good luck

David Shamouelian, MD
Facial Plastic Surgeon