Why does sulfur form S8 instead of S3?
Trisulfur does exist, but is not circular, it is ozone-like and unstable at common temperatures. 

My guess is that while angle strain does exist in ring form, it is not that harsh. Unlike carbon, having natural bond angle ~109 degree, for bivalent sulfur it is closer to 90 degree (sulfur exhibits little to no hybridization).  However, the S-S bond strength is only 40% of C-C bond and in trisulfur ring  all S-S bonds would have dihedral angle equal to zero, which is unfavorable because of nonbonding pairs interaction. This issue is a lot more severe than in tetraphosphorus, and leads to ring opening. 

For further information on sulfur allotropes see the Wikipedia article.