My best estimate on what would happen is the following:
- At the twilight zone there would be tremendously fierce and constant windstorms as the heavier cold air rushed from the dark side to displace the lighter air warmed by the Sun.
- Additionally, the wind patterns would generally align themselves towards and away from the twilight zone.
- Over a millennia, the storms would probably carve canyons and deep scars into the face of the planet along the twilight zone.
- Relative calm would be found in the area closest to the Sun as well as farthest from the Sun.
- If the planet were inhabited, the inhabitant would probably be locked to either the light side or the dark side unless they had some method of navigating through or around the storms.
- I don't know if it would rain or not, but if it did, it would definitely only rain on the side facing the Sun, as the other side would be perpetually frozen.
I'm not a meteorologist, this is just a thought exercise, so I don't know how accurate any of this is (although I do have some basic knowledge of meteorology).
EDIT: I found this wikipedia article, which seems to support my theories.
Due to differential heating, it was argued, a tidally locked planet would experience fierce winds blowing continually towards the night side with permanent torrential rain at the point directly facing the local star, the subsolar point. In the opinion of one author this makes complex life improbable. Plant life would have to adapt to the constant gale, for example by anchoring securely into the soil and sprouting long flexible leaves that do not snap. Animals would rely on infrared vision, as signaling by calls or scents would be difficult over the din of the planet-wide gale. Underwater life would, however, be protected from fierce winds and flares, and vast blooms of black photosynthetic plankton and algae could support the sea life.