Mostly these are the same things, and can be used interchangeably when casually talking about the option’s price.
But if you want to get technical, the option’s premium is what it’s trading for right now on one of the exchanges. This is the price where you can buy or sell it immediately.
Now, at the same time, you can have your own “value” of what you think the option is worth, based on your mathematical calculations.
Even though the option might be trading for $3 per contract on the exchange, your calculations might have it valued at $2.90 per contract.
The difference here comes down to the future expectations of the stock and what inputs are used to calculate the option’s price.
You may have different ideas than what the rest of the market thinks. Who’s right? You may never know, as it all depends on where that stock ends up on expiration day. Heck, you both could be right.