What's the difference between tamari and soy sauce?

I have a bottle of San-J Tamari (black label) in my fridge and the back reads:

Tamari is a premium soy sauce made with more soybeans than ordinary soy sauce giving it a richer, smoother, more complex taste. Tamari has more flavor enhancing properties than salt. Add 1 tsp. (320 mg sodium) instead of tsp. salt (590 mg sodium) to reduce sodium intake. Stir-fry or marinate poultry, meat fish and vegetables. Add 1-2 tsp. to perk up sauces, soups, gravies and casseroles.

In my experience, I use a little less tamari when substituting it for regular soy sauce. As Sean mentioned, it's definitely a bit stronger than regular soy sauce. I eyeball most of the time, but I'd say I use 1/2 to 3/4 portion of tamari when substituting it for regular soy sauce.

I reach for the tamari when I want a bit more of a complex flavor on something plain like rice - when the soy sauce is to be the star of the dish, essentially. I tend to use regular soy sauce when mixing into a larger homemade sauce or where the soy sauce flavor will blend into the background because the recipe calls for so many other strong ingredients.

Oh, and tamari seems to add a bit of an almost "smoky" flavor, it seems.

It doesn't answer the follow-up question, but one subtle difference is that tamari doesn't always have wheat in it, while soy always does.

This means, if you're cooking for people with gluten intollerance, some types of tamari is safe, while soy sauce never is.

(I've made the mistake of using soy when cooking for someone ... I now have a bottle of wheat free tamari stashed for the next time, but I've yet to compare the two directly)

Tamari is a particular Japanese variant of soy sauce. It's a bit stronger, though I'm not sure of the actual differences in production between the two. If you substituted standard soy sauce in for tamari, I'd imagine the recipe would taste less of the sauce, at a rate proportional to the amount of sauce for which the recipe calls. Maybe adding more soy sauce to the recipe would approximate the effect, or reducing it by some amount before adding, but that is pure conjecture on my part.

Tamari is a byproduct of making misu. It is the real deal. Shoyu is a tamari imitation made by altering the misu process to increase liquid production without hurting flavor. (It almost succeeds. not bad but not quite as good as the real thing.) Soy sauce could be either of these mixed with other fillers to increase production volume or yet another product designed to taste similar. I have seen both variants. When substituting tamari for soy sauce I normally use 1/2 tamari, 1/4 pineapple juice, 1/8 worcestershire sauce, a dash of ginger, a hint of tabasco sauce, and water to fill the volume if needed. Going the other way I just use extra and cut the water if needed.