Since they both describe that onsets take priority over of codas, what is the difference between them?
Minimal onset satisfaction allows the syllabification of a sequence like atra as at.ra, even when tr is a valid syllable onset, so long as r is a valid "minimal" syllable onset. If tr is a valid onset, it would be necessary to syllabify atra as a.tra when following a rule of onset maximization.
An example of minimal onset satisfaction can be found in the syllabification of prefixed words in Latin.
The following clusters all exist in word-initial position in Latin, and so can be inferred to be valid syllable onsets: /bl/, /br/, /dr/
Example: brevis "short"
Attaching the prepositional prefixes /ab/ /ad/ /ob/ /sub/ to vowel-initial bases creates words that scan with light initial syllables in poetry during the Classical period. This is theoretical evidence for the presence of resyllabification of the consonant to the onset of the following syllable in accordance with a rule of "minimal onset satisfaction" (which can alternatively be conceived as "avoiding empty onsets").
Example: sub- + ago, actus = su.bigo, su.bactus1
Attaching the same prepositional prefixes to bases starting with /l/ or /r/ creates words that scan with heavy initial syllables in poetry during the Classical period. (In the case of sub-, the coda consonant of the prefix may assimilate to a following r, resulting in a geminate /r.r/.)
Example: sub- + rogo = sub.rogo/sur.rogo
Since the second syllable already has a satisfactory minimal onset /r/, resyllabification does not apply, even though Latin phonotactics allow /br/ as a syllable onset.
Onset maximization would result instead in the syllabification *su.brogo with a syllable onset cluster br.
The scansion of the following line of dactylic hexameter shows that subactus starts with a light syllable, which is interpreted for Latin as a syllable ending in a short vowel:
tum vero adsurgunt irae, insidiisque subactus, (Vergil Aeneid line 495)