A first deposit offer is when you make a first deposit with a certain amount of money, and then the casino gives you some money back in the form of a bonus. In this case, the casino will give you some percentage of your total deposit amount that was used to make the first deposit.
A welcome bonus, on the other hand, is a bonus given out for making the first deposit at your chosen casino. It's essentially free money for playing on their platform. It's often called a "welcome bonus" because it's given to new players who've never played at that casino before.