Prices of a Future contract and a Forward contract are the same under the Black-Scholes assumptions (deterministic rates) but the price of options on Futures or options on Forwards might still differ. I did not find this obvious at first.
For example, when the underlying contract expiration date (Futures, Forward) is different from the option expiration date. For a Future Option, the Black-76 formula can be used, the discounting is done from the option expiry date, because one receives the cash on expiration due to the margin account. For a Forward Option, the discounting need to be done from the Forward contract expiry date.
European options prices will be the same when the underlying contract expiration date is the same as the option expiration date. However, this is not true for American options: the immediate exercise will need to be discounted to the Forward expiration date for a Forward underlying, not for a Future.