Difference Between 加 and 添 Do they both mean "to add", or is there some other context that I don't understand?

Thank you in advance!
You drive to a gas station to "加油".  We don't say "添油".

If you add some firewood to a campfire, we usually say "添柴".  If you say "加柴", if still makes good sense, but does not sound as natural as "添柴".

Otherwise, the character "添" is usually used with other characters, like "增添".

Although ""增添" and "增加" have the same meaning, they're not fully interchangeable.