Whether you use minimum or minimal depends on what you are trying to say.
minimum, n. and adj.
B. adj. (chiefly attrib.).
- [The noun used chiefly appositively.] That is a minimum; of or relating to a minimum; that is the least or lowest possible, usual, attainable, allowable, etc.
- Relating to or constituting a minimum; of a minimum amount, quantity, or degree. More generally: extremely small; very slight, negligible; constituting a bare minimum, only just adequate.
What this means in the real world is that minimum is the smallest amount, and as such usually takes a definite article - "we made the minimum change" - and is often part of a noun phrase. Minimal is small, negligable, sort of minimum-ish, and as such takes an indefinite or zero article - "we made a minimal change/minimal changes".
A nice example (or at least it works for me...) is residue in a wine bottle.
There was the minimum residue in the wine bottle ~ there could not be any less residue, which would suggest none at all
There was minimal residue in the wine bottle ~ there was so little residue, less than other bottle of wine, that it barely mattered at all.