Three terms get used loosely and they mean different things. Acclimatisation is the few days during which a cigar adjusts to a new environment. Rest is the few weeks to a few months that follow, during which the components settle into agreement with each other and any residual youth fades. Aging is the year and beyond, where the leaf begins to change character rather than just condition.
The cleanest way to think about it: acclimatisation fixes location, rest fixes condition, aging changes character.
Most cigars want the first two. Only certain cigars reward the third. A daily-driver Connecticut bundle is built to be smoked young and will not become something it is not after five years on a shelf. A Padrón 1964 Anniversary, an OpusX, an Aniversario, or a serious Cuban regional release is built from leaf with somewhere to go, and will reward five and ten years of patience with subtler edges, more integration, and more character.
If you bought a cigar to smoke this month, rest it. If you bought a box to lay down, age the surplus and smoke the working examples. Mark the boxes. Trust the calendar.
For the full argument, including the windows where each phase is doing its work, see How to Rest a Cigar.