Answer
Both are contracts of loan, but they differ in object and effect. In commodatum the lender delivers a non-consumable thing so that the borrower may use it for a time, free of charge, and return the very same thing; ownership stays with the lender and the contract is essentially gratuitous (Civil Code Articles 1933 and 1935). In mutuum — a simple loan — the lender delivers money or another consumable thing, ownership passes to the borrower, and the borrower must return an equal amount of the same kind and quality, not the identical thing (Articles 1933 and 1953).
A further consequence follows from this distinction: a commodatum is by nature gratuitous, whereas a mutuum may be with or without interest — but any interest must be expressly stipulated in writing to be demandable (Article 1956).