bon/meilleur v. bien/mieux

bon/bien

Bon (good) is an adjective. Although it has an irregular feminine form, bonne, the plural is formed regularly by adding  an -s to the masculine or feminine adjective. 

Bien (well, really, very) is an adverb. 

The adjective bon modifies a noun, whereas the adverb bien modifies verbs, adjectives, or other adverbs. 

meilleur/ mieux

Comparisons with bon and bien are not formed regularly using 'plus ... que' (more ... than). Instead use meilleur(e)(s) que, and mieux que: 

The adverb bien is an intensifier and can be translated as 'much' in English. Note that 'much better' is sometimes translated as bien meilleur (adjective), sometimes as bien mieux (adverb). 

le meilleur/ le mieux : superlative forms

In French as in English, the superlative is a form expressing a maximum or minimum quality or capacity.

In English,  'good' and 'well' have the same maximum superlative: 'the best'. 

In French, the maximum superlatives are formed with the definite articles le (la, les) meilleur(s) (superlative adjectives) or le mieux (superlative adverbs). The minimum superlatives (le moins bon, le moins bien) are formed regularly.