They are, in Japanese term, all subcategories of 笑い, the widest word that covers both loud laugh (giggle, chuckle etc.) and silent smile (grin, smirk, simper etc.)
The two words are related to every kind of silent happy face (abovementioned). 笑み is the action itself (< verb 笑む), but is a bookish word that hardly appears in conversation. 笑顔 is more popularly used one that refers to the face (or expression), and we usually call making smile 笑顔を作る.
Literally "dim happy face", may be the most exact words correspond to English smile. However, these are a bit more nuanced than the English counterpart where they're associated with affection, dearness or benevolence, that don't fit every situation we could use smile. The most typical usage would be 天使の微笑み "angel's smile".
微笑【びしょう】 is the Sino-Japanese synonym of 微笑【ほほえ】み, thus sounds loftier. Note that the last okurigana of 微笑み isn't obligatory, so sometimes you'll see 微笑 is read ほほえみ too.
This is somewhat an odd one out, and has most limited range of usage. It has two meanings: making a friendly happy face lightly/briefly as in 破顔一笑 "(break into) a broad smile"; and laugh something down/off as in 一笑に付す. Both are often seen in literary works.
This is a mimetic word depicts such a sunny smile:
for this kind of evil smile, we use にやにや or にやける (verb), which is frequently heard in anime when girls rebuke guys that have dirty mind.
and for big grin, we prefer にんまり.
We also use English loanword スマイル, usually for the cheerful smile or grin which Americans are particularly good at.
(for illustrative purposes only)