In arts|lang=en terms the difference between minimal and minimalism

is that minimal is (arts) characterised by the use of simple form or structures while minimalism is (arts) a style of art that emphasises extreme simplicity of form.

In music|lang=en terms the difference between minimal and minimalism

is that minimal is (music) characterised by the repetition and gradual alteration of short phrases while minimalism is (music) a style of music that emphasises extreme simplicity of rhythms and melodic forms to achieve a trancelike effect.

As an adjective minimal

is the smallest possible amount, quantity, or degree.

As a noun minimalism is

(arts) a style of art that emphasises extreme simplicity of form.