In most ways, these two analyses are very similar: both are gravimetric analyses, targeting hexane-extractable, semi-volatile hydrocarbons including any compounds present >C50. Extracts in both methods are treated with silica gel to remove non-polar (non-petroleum) compounds.
However, extracts analyzed using the F4 gravimetric method are dried in an oven at 100C for ~2 hours; in contrast, extracts analyzed using the Total Oil & Grease method (mineral portion) are dried at 40C in a fume hood. This means that samples analyzed using the Mineral Oil & Grease method would capture a higher proportion of hydrocarbons which would otherwise be lost at temperatures between 40C and 100C.