I love questions that I can actually contribute to. My answer will be focused solely on how US agricultural practices & environmental research changed as a direct result of the Dust Bowl. This also spans a bit to the side both geographically & chronologiclally as there were some other things happening concurrently in non-Dust Bowl parts of the US. I would contend that the long-term effects on agricultural practices are the main, national-level lasting effect of the Dust Bowl taken by itself instead of the Bowl as part of the wider Great Depression.
So, let's cover what caused the Dust Bowl in the first place. The 1920's saw a drought across much of the US agricultural heartland combined with rock bottom crop prices^5. This actually led to an increase in farmed acres as farmers were trying to expand their operations in order to grow enough to break even. However, the newly plowed farmland was generally not great land for growing things on. Think of fields that would be hard to get a tractor into, stuff that flooded regularly, etc. These new areas were farmed out of desperation & necessity , not because they were productive^4. Plowing techniques at the time were not terribly refined either. Contour plowing was not widely practiced or known by most farmers, let alone things like cover crops or strip till. I do not know exactly what sort of plowing equipment your average 1920s farm would have used, but after plowing I would imagine that the field looked something like a modern chisel-plowed field as chisel plowing still has 0 value for preventing erosion. Notice how exposed the soil is and the size of some of the soil clumps. This will be relevant later. On top of that, some soils are more prone to erosion than others, with sandy soils being the worst offenders^3. This was not really known at the time as a modern understanding of soil as a biome (instead of being chemically identical to crushed rocks) was not really established until Dokuchaev's work in the late 19th century. Crop rotations were not always used either and this caused problems both environmentally and economically. (My favorite thing out of this era is Boll Weevil Monument. I want to go see it so badly.)
We've established that the farming practices at the time did not generally allow for soil conservation/erosion prevention. Let's talk about the weather. Dry soil, such as in drought conditions, gets crumbly & breaks apart more easily than when moist. Additionally, plant root systems actually help keep soil in place (this the reason for using cover crops) and this was also impacted by the drought. We've got a combination of crumbly dry soil, minimal artificial efforts to keep the soil in place, weak root systems, and last but not least: the wind. The Great Plains & Midwest are known for being windy and most of the large scale soil erosion that happens out here is wind-based. Soil gets picked up and blown for miles. I've seen farmers around here plowing fields with dust clouds following the tractor. That's the setup for the Dust Bowl.
By this point, the economy is in the gutter and the Dust Bowl (round 1! It was actually a series of drought waves) was in full swing. FDR gets elected and the New Deal starts up in earnest. There were a whole bunch of programs aimed at farm aid. There were programs to buy surplus, plant trees, provide loans, build windbreaks, educate farmers, you name it. The federal response was huge and Congress got involved directly. Enter the Soil Conservation Service (SCS), stage right!
The SCS started out as an agency whose main question was "How do we stop this erosion?" and they started digging into researching the problem (heh). As I understand it, a lot of the boots-on-the-ground planting & construction work ended up done through the Civilian Conservation Corps (CCC), the main(?) New Deal program for improvements to public lands. Skipping ahead a few decades, let me just list off the things still in use that directly derive from the programs and research of the Dust Bowl Era:
The US Soil Survey. One of the first steps for figuring out how to prevent this level of erosion from happening was to understand the soils themselves. The data on soil properties get used for everything from siting buildings to digging wells, to installing septic systems, to designing drainage plans. The website & data are used daily by engineers, farmers, etc all over the country.
Soil conservation efforts became more widespread. This includes things like terraces, swales, crop rotation, cover crops, contour plowing, etc. This also includes research $$ funneled into figuring out how to do these things more efficiently. The research is possible in part because of the Universal Soil Loss Equation^1 (updated version: RUSLE2) which helped quantify various factors such as soil erodibility, water speed, and density of plant cover.
The existence of the USDA Natural Resource Conservation Service as the successor to the SCS. The purview of NRCS has widened from strictly soils and now also includes things like managing fertilizer runoff & developing small-scale wildlife habitat. There are also programs through the US Farm Bill which are aimed at farmers who want to address these issues on their own land.
The Conservation Reserve Program. CRP provides farmers with conservation advice & financial incentives to actively address erosion issues on their own farm. Enrolling in CRP also gives farmers a way to make money on marginal crop land instead of farming it & causing erosion problems for themselves down the road.
I would assume that living memories of the Dust Bowl helped propel the environmental movement of the 1970's but I do not have a source handy to support this claim.
The Dust Bowl kickstarted soil science as an established discipline in the US. Soil science as a discipline also concerns itself with the effects of pesticides, nutrient runoff, and has a large intersection with hydrology/engineering. For example, the Gulf of Mexico Dead Zone would be an even larger problem to tackle without the knowledge base that started developing as a result of the Dust Bowl.
So yeah, there's a lot that still happens in the agricultural, environmental, scientific, and political spheres that can be traced back to responses to the Dust Bowl. In these areas, it did not so much leave scarring as fundamentally shift the approach to various problems. The Depression in general saw a large reworking of the role of the federal government in general, but that is a larger topic better suited for someone with detailed knowledge of the New Deal.
If someone else has more insight into long term social effects, I will defer to them but as for the purely social aspects, the end of the Dust Bowl coincided with World War II. WW2 did a whole heck of a lot to pull us out of the Depression in general and I think the memory of the Dust Bowl's impact got largely absorbed into the cultural memory of the Depression as a whole. The main exception I can think of is John Steinbeck's novel "The Grapes of Wrath", which follows a family that was driven off their OK farm by the Dust Bowl. I suspect that this book is the best representation of a purely social-effect cultural memory of the Bowl in particular, as it's widely regarded and still taught in schools. Additionally, I'm sure that if you went to areas that were hit directly by the Bowl you would find evidence of communities that disappeared, farms sold/abandoned, and stories accompanying the same.
Further Reading:
edit - spelling, punctuation, clarity
