The best way to envision such a world is with a coordinate system such that the south pole is always facing the sun, the equator is locked in permanent twilight, and the north pole is locked in permanent darkness and ice. Lets consider three parts of climate: sunlight, air circulation (which controls precipitation), air temperature. The big thing we are not considering is arrangement of continents which will have a big effect on everything.
Sunlight
Let us assume that total average solar irradiance is about the same on Proxima b as on earth. In that case, the south-pole recieves ~2.5 times as much sunlight energy than the equator on earth, since it is always recieving sun directly overhead. I don't know how to embed equations, but the math on this is that sunlight incidence for a single spot on the equator lasts for 12 hours a day, and is a sine function, starting at magnitude 0 at t=0 time units, peaking at 1 at t=1 time units and going back to 0 at t=2 time units. The total sunlight can be modeled as the area of a half-circle of radius 1, or 1.57 units of sunlight. Constant radiation at magnitude 1 for 4 time units would be 4.0 units of sunlight.
At 30 degrees from the south pole (or 60 degrees south) the planets recieves 87%, at 30 degrees south 50% and at the equator 0% (also, obviously, the same on the dark side). Thus, everything below ~25 degrees south will get more more sunlight than the equator on earth, while above that point, sunlight rapidly drops to zero. The sunniest places on earth recieve ~8 kWh/day of sunlight, after counting for clouds, at the surface (those are deserts in the tropics). If we adjust the south pole of Proxima b to get a 10 kWh/m^2/day (which will be reduced by rainforest clouds to more like that of the earth's equator), then 60 degrees south sees 8.7 kWh/day (equivalent to Death valley in the summer), 30 degrees south sees 5 kWh/day (Chicago in the summer, or Miami year-round). At 10 degrees south gets 1.7 kWh/day (Seattle in the winter) which is barely enough to support plant life. Then all the sudden a the equator, the sunlight drops to zero. So there is a huge and sudden cutoff in sunlight energy.
The photoperiod is the amount of time per day the enviornment is exposed to the sun. This is 100% for the interesting half of the planet, and 0% for the not interesting half. This would have significant effects on plants. Alaskan cabbages can grow hugely because of constant sunlight, and anyone from the north (Minnesota/New England) can attest to the explosion of green that long summer days bring.
Air Circulation
On the earth, the Intertropical Convergence Zone follows the sun's zenith point back and forth between the two tropics with the seasons. Air flows from ~30 degrees north and south of this latitude towards it, and then rises, causing low pressure and rainfall. Since the equator is never too far from the ITCZ, it is almost always getting rain, and hence rainforests. On the Proxima b, the effect is magnified by the fact that the ITCZ never moves from the south pole, but the low pressure effect is somewhat disrupted since instead of expanding to a greater area as the air masses move towards the equator on earth, they air masses are compressed into a tighter spaces as they converge on a single point on Proxima b. On the other hand, the area of rising air will extend farther from the point south pole of Proxima b than it would from the line equator on earth.
Upwind, the Hadley cell will bring high pressures and lower rainfalls at 30 degrees off from the south pole, corresponding to 60 degrees south on our planet. Above that, the Ferrel cell will pull surface winds from 60 degrees south to 30 degrees south where that air will rise, and contribute to the Polar cell (note: badly named on Proxima b, since the polar cell actually concerns the equator here), that will suck surface winds from the twilight equator to 60 degrees south. The Ferrell cell, which is weak on earth, will likely be strengthened on Proxima b, since the sunlight is constant in the various parts of the planet. However, the precipitation effects of these cells would not be seasonal at all, since there are no seasons, so there will be no rainfall regimes equivalent to the summer-wet Savannahs or winter-wet Mediterranean climates of earth.
Air Temperature
The division of Proxima into circulation zones will have some interesting effects.
The Hadley cell will recirculate warm/hot air from 60 degrees south to the south pole. The (adiabatic) compression as the air approaches the point of the south pole will probably result in higher temperatures, so that the rainforest center will be even hotter than expected. Rainforests have an average yearly temp around 26C on Earth, so this on Proxima b they might be more like 32C, the yearly temp in the deserts of Djibouti (total guess, no math involved).
The Ferrell cell will bring the still warm/hot air temps from 60 degrees south up to 30 degrees south. The polar cell, on the other hand, will bring cold/frigid air from the equator to 30 degrees south. I would expect this to cause a sharp temperature grade at 30 degrees south.
Put it all together
At the south pole, and for ~20 degrees latitude (1200 miles/2000km) outwards is a rainforest, with constant clouds rain, and heat. On an earth-size planet this rainforest would be ~12 million km^2, bigger than China and more than twice the size of the Amazon, assuming it was all covered by land. The heat would increase towards the south pole so that the pole itself would be unbearably hot and humid for a human, or any sort of warm-blooded creature (assuming earth-like biology). This are would be a haven for insects, reptiles (Dinosaurs!) and amphibians.
At this point the precipitation levels would drop steadily into barren conditions at the 60 degree south. The trees will gradually transition to evergreens that are able to retain water better. As he dense foliage thins out, eventually we reach a point where this is enough underbrush to burn periodically during week long dry spells. Then the trees have to be fire resistant and the ecosystem would resemble the evergreen thorn forests of India. The next step would be hot semi-desert like Arizona with its giant cacti, and then finally sandy dunes like the Sahara.
Going from the dry zone back to the wet would be similar. Because sunlight energy is still high here, and warm winds are coming from the 60 degrees south belt, temperatures stay high between 60 and 30 degrees south. I mentioned that solar insolation at 30 degrees south was like Chicago in summer or Miami year round. Well Chicago in the summer is 24C, and Miami is 25C year round. So its still plenty warm around here. Infact, this region would be another rainforest, just like the tropical one, with constant moisture warm temperatures. However, with the reduced sunlight, it might favor coniferous trees over broad-leaved ones. Lower sunlight means slower growth rates, which might favor trees that can grow higher. The biggest trees on earth are conifers.
After this there is a sharp temperature cutoff as warm dy winds rise over cold polar ones coming from the north. This sounds like a very stormy zone, with temperature fluctuations from hot to cold as fronts move back and forth, and massive thunderstorms. It should still be wet, but the trees must be adapted for periodic freezing temperatures.
Then above the transition zone, the winds would bring nothing but frozen moisture-less polar air. Since there is still some sunlight, and it is constant, the soil should not freeze even if the air is frozen, and plants that can grow in near freezing conditions can still thrive, like mosses and lichens. This area would be an evergreen but cold tundra with temperatures oscillating around teh freezing point.
After crossing the equator, the sunlight suddenly drops off to nothing and life ceases. The seas are frozen, and the lands buried in ice.
