It certainly means "a great deal more" but from the original term in mathematics, it actually means "many times more" easy. Three times easier, five times, ten times easier. "A great deal" would be more additive than exponential.
Rabbits when considering parenthood are said to adopt an exponential principle of a kind - 2, 4, 8, 16, 32, 64, 128, 256, 512 etc. That way they quickly come to have a lot more rabbits. An exponential graph is shaped like the horn of a trumpet.
This number sequence is a geometric progression. Many mathematicians will tell you that there are important differences between geometric progressions and exponential functions. Many word people don't differentiate between them, I suspect. The key point for us here is that the increase at each stage is increasingly dramatic, and, if the stages are close together in time, we become overrun with rabbits.
For anyone who has been involved with mathematics, the difficulty with using exponentially in this way is that it has no meaning.
Things may increase exponentially, referring to the rate of change over time (for example). But even then, exponential growth could be either extremely slow at first, gradually becoming faster and faster, until the bend in the curve is reached and the growth becomes extremely rapid extremely quickly. Curve one.
Alternatively, exponential growth could be extremely rapid at first, gradually becoming slower and slower. Curve two.
However, having got that out of the way I can accept the general sense that exponentially greater means a lot more. What I would resist is any attempt to quantify it