From AIRBUS:
EXPEDITE CLB aims at reaching the best climb gradient - which is driven by Greendot, S or F depending on the configuration.
Benefits will vary according to the gain expected:
Short term: If the aim is to get a better climb gradient for the next 10 or 20 seconds (avoiding manoeuvre for example), selecting the EXPEDITE mode allows to take benefit of the transfer of energy from kinetic to potential and of the speed closer to the speed of best climb gradient.
Long term: If the aim is to reach a certain altitude as quickly as possible (ATC demand for example), the gain in selecting the EXPEDITE mode before flaps/slats retraction is negligible compared to a normal climb (without expedite).
Conclusion: Accelerating normally the aircraft in order to reach as quickly as possible the clean configuration, and selecting the expedite mode once the green dot (best lift to drag for the clean configuration) is available is a good compromise between acceleration and climb gradient. In addition, this provides more quickly more energy and better defence should an engine failure occur for example.