Tilburg has a standard northwest European climate — wet, mild, and grey for most of the year, with the best weather in May, June, and September. July and August are warmest but often humid. The city's main reasons to time a visit around a specific date are Carnival and Roadburn.
Carnival (late February or early March, three days before Ash Wednesday) is Tilburg's biggest annual event. The city goes fully into it in a way that most of the Netherlands does not. Street parades, brass bands, fancy dress, and packed bars from morning until well after midnight. It is a genuine local tradition, not a tourist performance, and the energy is different from anything else in the Netherlands outside of the southern provinces.
Roadburn Festival (April) draws an international crowd of heavy and experimental music fans to 013 and several other city venues over four days. If that's your world, it is one of the best festivals of its kind in Europe.
For outdoor activities, late April and May are best — the cycling routes through the Loonse en Drunense Duinen dune landscape and across the Regte Heide heathland are at their finest in spring and early autumn.