Smile, grin, simper, smirk are comparable as verbs meaning to express amusement or pleasure or satisfaction or, sometimes, contempt or indulgence, by a brightening of the eyes and an upward curving of the corners of the mouth and as nouns denoting such an expression.
Smile is the general term, capable of being qualified so as to suggest malign as well as benign pleasure or amusement.
Grin implies a broad smile that shows the teeth. It often carries some suggestion of grimacing in anger or pain in its not infrequent implication of unnaturalness, of bewilderment, or senselessness.
More often, however, grin tends to imply naïve cheerfulness, mirth, or impishness.
Simper implies a silly, affected, or languishing smile.
Smirk , too, suggests affectation together with self-conscious complacency or embarrassment.