Yes, it is a different language. It is built for expressing ideas about deduction and quantity clearly, flexibly, creatively, and it works beautifully. Saying 2^3 = 8 and log_2 8 = 3 sort of get at the same fact but not really, that's misunderstanding their purpose. They express that we are evaluating functions. You can express 8 - 2 = 6 or 6 + 2 = 8 "in the same way" (Crazy! How can we have two notations +, - when they are just inverses of each other?! We shouldn't give ourselves language to express both "the difference between 8 and 2 is 6", and "2 more than 6 is 8" because they happen to be rearrangements of the same equation!) but the equations are used to convey meaning in different ways in different contexts.
The second example, the 8^(1/3) is not even equal to 2, it's equal to three values, two of them are imaginary. It's important to have notation for "the thing that when you cube it is equal to 8" distinct and understood so that when you begin understanding imaginary numbers (high school iirc) you have context. Then you can explain the definite article in that quoted sentence is actually inaccurate. What if he had selected an example with two real roots, like sqrt(4)? What should we put in the the "triangle of power"? The positive branch cut? Okay, now you have to explain that you're really doing a different operation now, that has multiple answers, but we are going to pick one of them and put it there, but we have to remember that it could be either. Which is best expressed using separate notation to explain the operation you are doing that isn't even a function.
The fact of the matter is that these ideas are distinct and relating them is a separate, worthwhile exercise that helps understand the structure of exponentiation and its inverses.
And in doing mathematics you will find that switching to equivalent but more informative or clean or applicable notation is one of the most valuable workhorses we have for solving simple problems.