Like most tidally locked planets, there will be a "hot pole" facing the heat source (in this case the Brown Dwarf) and a "cold pole" facing space. These two poles will be the drivers of much of the planetary climate, as the atmosphere and water vapor is heated and expands away from the hot pole and moves in large convective cells towards the cold pole, where it sinks and heads back towards the hot pole. So there will be a constant wind at ground level blowing towards the hot pole and a "jet stream" like wind system moving from the hot pole towards the cold pole. There will be a more or less solid ring of clouds around the hot pole as water vapour condenses out at higher altitudes, and possibly another ring of clouds somewhere on the night side as remaining vapour condenses out in the cold temperatures, so there will be two planetary "belts" with high levels of precipitation; rain around the hot pole and snow around the cold pole.
In terms of ecospheres, you could think of the planet as a beach ball with a series of concentric rings radiating away from the hot pole to the cold pole, and the ecology is determined by the varying amounts of energy and moisture being received in each "ring". The distant sun will provide a small amount of energy and illumination, although if it is as far as you say, this could be minimal (maybe even as little as a full moon on Earth). Because the planet would be orbiting the Brown Dwarf fairly closely, the illumination would also be rather sporadic, with the distant sun rising and setting quite quickly. Most native life would have to be adapted to "seeing" and harvesting energy in the infrared band, so plants would probably be black in colour and the eyes of native creatures would be very large to gather enough light, to begin with, and have the proper adaptations to receive and image infrared wavelengths, so the eyes would look different from ours.
Although you have not mentioned other moons of the Brown Dwarf, it is quite likely that they would exist due to the intense gravitational field. Multiple moons would probably settle into resonant orbits (moons in nonresonant orbits get "pumped up" with energy and change orbits, either being expelled, absorbed by the Brown Dwarf or settling in stable resonant orbits. This should provide an extra source of energy for the moon's various tectonic, hydrological and atmospheric cycles, as the core is "kneaded" by the interaction with the other moons and heated more than it otherwise might be. This internal heat is probably what is keeping the oceans liquid (much like the internal oceans on Europa and other Jovian moons).
The last thing which might affect the moon is the presence of a powerful magnetic field around the Brown Dwarf. Jupiter's magnetosphere provides a great deal of energy to the environment, and if the moon in your system is at the right distance, it might be interacting with the magnetosphere much like the Jovian moon Io, which is working like an armature in a dynamo and creates a multimillion amp "flux tube" between itself and Jupiter. Something like that would certainly cause a great deal of disruption to any planetary atmospheres, as well as make scientific discovery interesting (you might discover electricity early, but the roar of noise on radio frequencies will prevent the development of radio broadcast technology and radio astronomy, for example).