"Importance" is a fairly overused word in DL. e.g. importance sampling and importance weighted gradients.

"Attention" works by creating inductive bias for the upstream network, which is analogous to human attention, and the word itself is much more intuitive.

Keep in mind machine learning is largely a descriptive science(modeling the behavior), whereas neuroscience is more prescriptive. So from the behavioral perspective, attention is better suited than importance.