I think in project life cycle there cannot be any single phase that can be more important than other because the end project would be a result of proper research and approach (i.e is initiation phase) well developed plan, thoughtful execution of the plan and timely monitoring and bringing changes when and the very end proper analysis of the whole project, what went wrong, where it went wrong, and make sure everyone is taken care of there needs (i.e. closing phase) mark the end of a successful project cycle.

If I had to select the most important phase I would say it would be the current phase or the ongoing phase.Let take random example of planning phase while planning if you miscalculate the budget or set deadline to tight to achieve or improper communication planing etc all this would be reflected in your execution phase and would also affect monitoring and control phase.
Now we take other example of initialization phase if the manager forget to take into consideration few features or few specs which might add to the value of the device or if some information where left to assumptions all this might later come up during the project and would affect the overall cycle. So no matter what if appropriate focus, attention and care is not given on the current cycle it is gonna affect all other task and ultimately on the project cycle. So I think ‘CURRENT PHASE’ is the most important phase .