There is an old fishing saying that every time you go fishing the conditions will be different. Things are very different if the weather is cold. On the one hand you the fisherman will be working under different conditions but at the same time the trout will react differently.
From the fisherman's point of view the less distractions you have the better. If you are freezing or as happens quite often, the longer you are out the more you feel numb, then it is difficult to cast, to keep your momentum and your mind will certainly turn to your physical sensations. Perhaps the best cold weather tip ever is that you lose warmth through your extremities, so make sure you do not lose your body temperature. A good hat or cap to keep your forehead warm and if it is extreme, use ear mufflers. Your feet are another area from which you lose heat. Good woolen socks, maybe two pairs, are a great solution. One of the advantages is that if you do get wet, in a very short while you will find the water in your boot gets warm thanks to the wool properties. Next, and sometimes the most important, is your hands. When fishing you have to be able to feel the line and if you use gloves you will lose feeling. The answer is fingerless gloves (woolen if possible). All the other clothes you can use, whatever combination and number of items you decide on, are important but secondary. Just make sure you keep your mobility. There is another anglers saying and that is i you do not get wet you wont catch anything. Wet and cold are not the most comfortable combinations, but wool is a godsend.
Now for the trout. The cold weather will influence the water that makes up the stream. The colder it gets the colder the water. And cold water makes for sluggish trout. Fish are cold blooded so the temperature of the water has a direct relationship with their blood temperature. You have probably seen reptiles (also cold blooded) that early in the morning hardly move at all. Later on you will see them sun bathing until their blood temperature rises and they then begin to move. A cold fish is a fish that will not be able to move quickly. This also means that cold fish eat less as they have not the energy to rush around hunting all over the place.
What can you learn from this? Read the stream so you can identify the likeliest place where trout will be resting. In the small pond, behind hidden rocks and by the banks with slow moving water. This is when casting becomes even more important. Cast as accurately as possible. If you are spinning and the lure lands with a great splash in front of the trout you will chase it away. If however you allow the current to softly take in front of him as naturally as possible, your chances will increase. The same goes with fly fishing. Do not expect the trout to chase after a fly or a lure, but serve it on a metaphorical platter.
Finally one of the advantages of cold weather is that not as many anglers will be around and this gives you a pleasant advantage.