Odense is at its best from late April to early September, when the King's Garden is in leaf, the riverside paths are busy, and the open-air sections of the Funen Village are in full swing. The Hans Christian Andersen Festival in late August fills the streets with theatre and music, much of it free, and is the single best week to time a visit.
Winters are mild for the latitude — temperatures usually above freezing — but daylight is short and the riverside walks lose much of their appeal. December does bring the cathedral's Christmas concerts and a small market on Flakhaven. Avoid the late February school holidays unless you specifically want school groups in the museums.