Bad weather is often described as a problem.
For photography, it can be the opposite.
Rain Changes Light
Cloud cover creates a giant softbox.
Direct sunlight disappears and contrast can drop dramatically.
Rain also creates reflections, texture and atmosphere.
The obvious approach is to stay inside.
The more interesting approach is to photograph what the weather changed.
Fog Simplifies Everything
Fog removes background information.
That can be extremely useful.
A complicated landscape can become a simple arrangement of shapes.
A city can become layers of light.
A person can become a silhouette surrounded by empty space.
Protect the Equipment
Weather photography is only fun until water gets inside something expensive.
Use appropriate protection, keep cloths available and be careful with temperature changes that cause condensation.
The camera is replaceable.
Your enthusiasm for photography may not survive destroying it.
What Actually Matters
Bad weather gives you conditions that do not exist on normal days.
Instead of asking whether the weather is good for photography, ask what the weather has changed.
That is usually where the photograph is.