You ask a great question. I have to give you some background to explain this issue. Because Judaism is fragmented and there are many different denominations different rules and understandings of the mandates arise. The best area I can speak to is what the laws are and then individuals are free to chose what they want to observe.
Kosher wine is defined as wine under the supervision of an approved "Hashgacha" agency. A Hashgacha is an agency that trains Rabbis to watch the production process. Inherently there is nothing unkosher about most wines. However just like there is nothing inherently unkosher about orange juice, unless it is marked as being kosher and has been certified by a reputable agency, observant Jews will not consume it. Well known agencies that certify things as kosher are the OU- orthodox union, the Star-K, Kaf-K as well as many others.
Now with that said there is another layer to make it more complicated. That is the rule of Mevushal. This concept is that the wine is "cooked" or "boiled".
Traditionally, this edict was followed literally. The boiling process killed most of the fine mold or "must" on the grapes, and greatly altered the tannins and flavours of the wine. The result was typically a weak wine. Rather than being full red in colour, it often displayed an opaque, permanganate-coloured tone.
Later, the process was modified to require only that wine be heated to 90 degrees Celsius. This managed to reduce some of the damage done to the wine, but still had a substantial effect on flavour.
Recently, a process called flash pasteurization has become the new style. This method avoids causing the juice of the grapes to simmer or boil, and is said to have a minimal effect on flavour, at least to the casual wine drinker. Indeed, the non-kosher winery Château Beaucastel flash pasteurizes and its wines are considered among the world's finest, although few others have copied this technique.
In most territories, the bulk of kosher wine is supplied by wineries producing both kosher wine and wine for the general market. However, irrespective of the method, the pasteurization process must be overseen by mashgichim (people certified in to rule on the Kashrut "Kosherness" of a product) to ensure the kosher status of the wine. Generally, they will attend the winery to physically tip the fruit into the crush, and operate the pasteurisation equipment. Once the wine emerges from the process, it can be handled and aged in the normal fashion.
There are Rabbinical reasons why these laws exist. I can go into them if you want. But the fact is that observant Jews hold by this idea of Kosher wine, (which is either mevushal or not mevushal - a wine does not have to undergo the mevushal process to be considered kosher).
Does this help explain your question? In essence there is no production difference (unless its mevushal), its a matter of who oversees the process. As I mentioned above different denominations practice these rules to varying extents.

There are some famous Champagnes that are Kosher, I thought. I just can't remember which ones...

Wine is a more manufactured product than, say, orange juice, so there are two areas in which a Kosher wine can intrinically differ from a non-Kosher wine, and one is in the area of what material is used in the filtering process, or "fining". European wines may use casein (milk protein), isinglass (a pure gelatin from fish swim-bladders, not urinary bladders!), or, rarely, beef gelatin. No kosher wines do. All Israeli and U.S. kosher wines filter only with purified clay (bentonite), making them vegan. So do most European kosher wines, though some may use egg albumin, a common practice for non-Kosher wine, so they are all vegetarian, but not reliably vegan.

The second is in the area of traditional taste profile. At the lowest price points, Kosher wine is usually sweet, sometimes syrupy, for Jews accustomed to traditional sacramental wines. Even some lower-priced Syrahs and Merlots have been slightly sweetened. However, some South American wines have been coming in a very low price point made in the more standard food-friendly dry style. Since some labels, like Herzog, include examples of both types, and reviews that describe the flavor profile don't always mention that the wine may be off-dry unlike most wines of the varietal, caution is necessary.

At the medium ~$20 price point there are quite a few excellent Israeli wines (where the cost differential of producing kosher is less than elsewhere) that compare favorably value-wise with non-kosher wines. Daniel Rogov has reviewed many of these.

From what I've read above, the thing that flags my concerns most when it comes to the quality, esp. complexity and longevity, of properly kosher wines would be the requirement for boiling (or even flash pasteurization) of the wine during the fermentation process.

Originally trained as an anthropologist, I have tended to view religious proscriptions regarding diet and certain other areas of behavior as most often stemming from environmental threats to the health and social hygiene of the particular historical communities, which have then taken on other significances over time. Easy examples are bans on pork (trichinosis and other diseases) or mollusks (numerous toxins that occur seasonally or in shellfish that are not fresh).

I'm curious about what health-related problems may have ensued from unboiled wine in the past?