What evidence shows that hydrogen reacts differently in water than it does in hydrochloric acid?
Your question could be interpreted in two different ways. If you mean taking hydrogen gas and bubbling it through water and hydrochloric acid and looking for a chemical reaction, I don't think that hydrogen gas (H2) will actually react with either of them. Hydrogen isn't even particularly soluble in water, and HCl plus H2 I don't think would give any reaction.
I suspect, however, that you are talking about the reactivity of hydrogen atoms in water (H2O) versus hydrochloric acid (HCl). The reason for the difference between the two has to do with the type of bonding hydrogen has in each case. In water, the hydrogen is covalently bound to oxygen, thus the water molecule as a whole is a neutral species. As such, the hydrogen in water is largely unreactive. Only when treated with a very strong base or an electrical current will the hydrogen in water undergo a chemical reaction. The hydrogen in HCl, however, is an ion. HCl is composed of the ions H+ and Cl- that are bound together via an ionic bond. In water, the HCl will completely disassociate into its constituent ions. The H+ ion (also called a proton) is extremely reactive and will not only react with any kind of base added but will also react with other types of functional groups as well.
The only evidence you need that the two are different is to test the pH of both chemicals. If you dip a pH meter into regular water, you will get a pH reading of approximately 7 (neutral). If you dip a pH meter into an HCl solution, it will read well below 7 (acidic).