How to distinguish cream of tartar from bread improver

This depends on what is in your bread improver. Cream of tartar is salt which acts as a buffer. If the bread improver also contains acid (ascorbic acid is sometimes an ingredient), it makes it harder to distinguish, because it will react similarly in many circumstances.

The first simple test would be to add baking soda to a solution of each ingredient. If only one sample fizzes, this is the cream of tartar, and the bread improver has no acid. If both fizz, the bread improver has acid and you cannot tell from this test.

The second test needs six jars of water (or smaller things, like espresso cups) and pH strips. Dissolve each of the ingredients in two jars, and fill pure water into the third pair. Add some baking soda to one of the water jars, and a colorless acid to the other one. Use a pH strip to see that you added enough to be able to measure it, a change in pH by 1 step should be enough. Then add the same amount of soda to a cup with ingredient A and a cup of ingredient B. Repeat with the same amount of acid. Measure the acidity of the new cups. The ingredient whose solution remained at a neutral pH is the cream of tartar. There is a small chance that both will do this, because we can't know if the dough improver also have a buffering quality, but I think this is not so likely.

A third test, if you don't have pH strips for the second: whip an eggwhite with a pinch of each ingredient. The more stable foam will be the cream of tartar. With some luck, the bread improver eggwhite will whip even worse than pure eggwhite, if the dough improver has emulsifiers (intended to soften the bread). But this test is not perfect either, because ascorbic acid or other ingredients used to make stronger bread could also stabilize the eggwhite - they are intended to strengthen gluten, but will work on eggwhite proteins too.

The best test would be to use not chemical behavior, but taste. For this, you need a new, labeled sample of either cream of tartar or the same brand of dough improver. You can just dip your finger in it and lick. Then repeat (with a different finger) in each of the unlabeled ingredients. As long as you are sure that these two are indeed food additives (you don't have an unlabeled box of NaOH sitting in your cupboard, for example), there is nothing dangerous about the test. The taste should be distinctive enough.