I'll preface this by saying in a single week I have dived in extremely warm Bahamian waters and then under ice in a New Hampshire lake.
I understand extremes of diving.
The simple part of the answer to the OP is warm water diving is relatively easy. Thermal protection, ie., wetsuits, are usually three of five milimeters(add a 5mm core warmer if I'm diving for an extended period). I normally wear something like six or eight pounds of weight. My backplate is aluminum and a single AL80 seems to last a long time. I generally use the same regs in warm or cold water, either AquaLung Legend Glacia regs or APEKS XTX 100 regs --they are enviornmentally sealed so they are great for extreme cold, but are equally perfect in tropical waters.
Continuing with warm waters, I use a lighter knife because the visibility is better and potential entanglements are easier to see. I don't wear a hood in warm water and if I wear gloves they are for protection from stings, not for warmth.
I do carry a SMB and small finger spool in warm water. Usually I don't carry a light but that varies a bit depending on the dive. Computers are the same for cold or warm dives, either VR3s (if I need redundancy) or the increcibly excellent LiquiVision Computer and usually my trusty Citizen Eco-Drive Watch/Computer. I always wear a wrist mounted compass.
I've tried pretty much every kind of fin around but my favorite is the APS Mantaray --I've used these fins with single tanks, doubles, side-mount in everything from glass smooth water to rolling surf and they are absolutely terrific. They are light, easy on the legs and give truly remarkable power with every type of kick I use.
The more difficult part of the answer is cold water diving.
I dive a DUI CLX450 drysuit in truly cold water, otherwise a BARE 7mm wetsuit with a
5mm hooded core warmer. For the drysuit, I recently switched to a Weezle undergarment, which is great. XS Scuba dry gloves, DUI hood and neoprene socks backed up with wool socks.
In colder water I use more weight, of course, which I distrubute on a weight belt and in other places to ensure good trim. I prefer a stainless steel backplate for cold water, although that seems to attract cold. I usually weight the lower part of the plate for trim.
While I do a lot of open circuit diving, using my KISS Classic rebreather in cold water really helps keep me warm.
I normally add ten feet of depth to my profiles to account for colder water temps. My dives tend to be more conservative. I always carry two lights, even in summer because at 20 or 30 feet, it starts to get dark. I have a reel with me, as well as a lift bag and spool and two cutting devices, one of which is a very robust knife.
But the real difference is psychhological. I love diving in warm water like nearly all divers. It's easy and pretty much every dive just seems great. But because I live in Maine in the northeastern US, a lot of my dives are in cold water; even in warm months by the standards of many divers. In winter, getting into the water even in a drysuit with a very thick undergarment requires nerve and mental focus. You have to be prepared, whether it is back rolling off a boat or wading in from the shore or slipping into a lake from a hole cut through the ice. That is absolutely the greatest difference between warm and cold water diving in my mind.
Jeff