maximum vs maximal

maximum is a noun but maximal is not a noun.

Word	Noun	Adjective	Verb	Adverb
maximum	Yes	Yes	No	No
maximal	No	Yes	No	No

As adjectives, maximum and maximal are synonyms defined as:

maximum (noun)	maximal (noun)
the greatest possible degree
the point on a curve where the tangent changes from positive on the left to negative on the right
the largest possible quantity

maximum (adjective)	maximal (adjective)
the greatest or most complete or best possible	the greatest or most complete or best possible