The biggest impact on technology would be less internal diversity within a region but more diversity across the globe based on climate areas. Let's take the areas you've identified already and expand on them:
A harsh year-round climate would lead to little life being able to develop. Life that does develop would be very slow-growing because the cold would slow down biological processes. This would result in little to no food for people to eat or animals to hunt, so chances are people would just avoid it. Edit: This would probably cause many rumors to be spread about treasure/artifacts/monsters/etc that live there and might cause someone wealthy to mount expeditions to find it. You'd end up with some hair-brained tech developed for the purpose that could fail horribly or succeed brilliantly. Just look at our earliest theories on flying machines for inspiration on how crazy it can get before we actually understand how it works.
These would be similar to the grassy regions I described in the other question. Grass-like crops would probably bud year-round and be difficult to harvest without hand-picking or some creative method of controlled cultivation. Simple wind and animal transfer of seeds into a field could disrupt its uniformity and require constant maintenance to ensure crops are harvested with some consistency. The lack of a definite growing season would also disrupt the natural budding cycles of these plants so some might mature faster than others. Basically farming grains would be a pain, so people in these areas would likely resort to a more nomadic hunter-gatherer lifestyle, unless there's an abundance of high-value food sources like fruit and game that prevent the need to move. Large populations would be very difficult to support either way. People would also likely get very territorial due to the need for a fine balance in supply and demand. Expect long-range weaponry due to wide-opens plains giving good visibility, or if you want something more unique, make the people very adept at stealth and camouflage and the short-throw spear or miniature poisoned bow. There are tribes in Africa that use these techniques to stalk wild animals in the plains, though their lifestyle is a harsh one indeed.
Deserts not near the equator could in theory operate much as they do in the Southwestern US. Mountains could easily divert rainfall periodically to keep plant and animal populations alive, and the population would probably rely on similar cattle farming methods to what you see in our world's history. Water-storing plants like cacti would be common. Strangely enough you might see this being one of the more prosperous zones, though populations would still be small. Expect lots of water collection technology to be a necessity, and if you want to make it harsher you could make water a precious resource, though that precludes the region being prosperous. See Dune for examples of how that could shape society.
Rocky Mountainous Regions
High up the mountains would be pretty much frozen and uninhabitable. In the middle altitudes you could have large lakes, though they would have to be fueled entirely by rainfall since you won't have spring melts of snow caps. I'd expect much of the population to center around the lakes and their distribution systems, and the people would live primarily off of hunting and whatever high-value fruit/vegetables can grow in the rocks. Expect small populations but relatively stable places to live since they have a good supply of water, which perpetuates more lush plant and animal life. Think of a "mountain oasis". I'd expect those people to be pretty reverent towards their water source as the giver of life. You might see an abundance of mining technology just out of necessity to build habitats, though if the mountain is very geologically active that could be a risk. If not, they could take advantage of the increase in temperature you get as depth increases, though be careful to avoid asphyxiation.
Swamplands and Rainforests
These would probably be your centers of civilization. The abundance of water and warm temperatures would mean an abundance of wildlife. People here would have wood to make tools and shelters, a huge variety of food to eat, possible easy travel down rivers, etc. Houses would likely be elevated somewhat to deal with flooding or in the case of the swamp they may just be permanently floating. You'd possibly have some sort of agriculture in the swamp regions, though it wouldn't be grains, perhaps edible reeds or other water-borne vegetables. In the rainforests you could also have good mining locations for metals and stone. A large enough population could clear out an entire swath of land to build an empire, though they would have to be mindful of the ground they build it on. Stone buildings would be useful, and wood structures would need to be treated with some sort of tar-like substance to prevent rot. These would also consequently be dangerous places to live, with the abundance of wildlife meaning there are just more things out there to kill you in strange ways, necessitating many precautionary measures and weapons/tools that would border on being ritualistic after several generations. Get creative!
Another potential for lush wildlife but only if they're in a warm enough climate to have lots of rainfall. Either that or your oceans would need to be freshwater and I'm not sure what the implications of that are. These places would probably be much as you see them on Earth, with tribal villages being the norm and relative isolation from the rest of the world until someone invades. You could build them up to be a culture clash after that happens, just think about occupation-era Japan for inspiration on how two cultures can combine in a totally unique way (not to mention the social stresses it would cause). Expect tools to be somewhat sophisticated but shelters to be impermanent due to the likelihood of being destroyed in a storm and needing to be rebuilt easily. I think it would be fun to have a culture that reveres the cycle of death and rebirth in these situations, as exemplified by the potential for weather to wreck everything, forcing them to rebuild from scratch. You could expand that into their whole style of technology, perhaps their tools are all one-use and disposable despite them knowing how to make better ones, they just choose to let things break as a sign of acceptance that they themselves may be broken on any given day that nature determines. Outsiders would view them as incredibly primitive when in fact they're incredibly intelligent and sophisticated, they just show it differently.
Other Interesting Bits
You can get creative with some areas like the hot spring you mentioned. People would probably have to have migrated there at some point, rather than it being a place of origination. The hot spring in a frozen tundra would support a unique kind of plant and animal life that could be very different from everywhere else, but the limiting factor is the size of the spring. The smaller it is, the less life it can support. If you make it a larger system of springs connected by short distances of streams (such that they wouldn't freeze over in transit from one spring to the next) you could extend it a bit, but remember this is going to basically be on a fault line so there will probably be mountains and other geological features surrounding it.
Just keep in mind that mountains are great at diverting rainfall, so you can bring water to places it wouldn't normally be, and since there would be a reduced ability to transfer water through natural means like seasonal melts, irrigation will be key to any developing area that doesn't have lots of water already. There could be an empire in the grasslands, but only if they have the water and trade supplies to support it. It would be much easier to build that empire in a wetter climate.
Also take advantage of massive rivers like the Nile or Amazon that connect two large bodies of water. The Nile in particular made life possible in Egypt, and that river alone allowed them to build an impressive empire despite the harsh environment, especially with the advent of river shipping techniques. A more advanced society with slightly different terrain might take advantage of the water to power massive mining operations or other large industry. As happened in history, these people would likely view the river as the source of life and it would feature prominently in their religion and tradition.
Wherever people live, providing shade would probably be an important feature of their technology. The lack of seasons would make constant sunlight harmful in many areas not near the poles. Consider large-brimmed hats, parasols, covered carts and full cloaks/robes as well as shaded places to rest an important feature between the grasslands and the equator. Good ventilation in any warmer area is also a necessity. Also make sure your distribution of skin pigmentation reflects the climate, assuming your people are biologically tuned like us.
Hopefully that covers it. If I missed something let me know.