Are there metals that react with sulfuric acid but not hydrochloric (or vice versa)?
OK, let's clean things up.

Reaction of an acid with a metal is generally a redox reaction. The oxidizing agent are protons generated in the process of dissociation of the acid (let's forget about anions for a sec). The property that decides whether a metallic element can be oxidized by protons is its standard reduction potential. In general, any metal with a negative reduction potential will react with any Brønsted acid (one that produces protons), no matter whether it is a weak (e.g., acetic) or strong acid (e.g., hydrochloric). And if a metal has a positive reduction potential, it won't react with acids. Therefore, if we do not consider other factors, there are no examples to make the point that different acids react differently with a given metal, in fact they all should behave in the same way.

But what makes Chemistry fun are those 'other factors'. The anions produced in the dissociation process of an acid can also act as an oxidizer. For example, nitrate ions produced from nitric acid will oxidize copper, so the metal will dissolve in nitric acid. Sulfate ions are also oxidizing agents, but much weaker than nitrate ions, so copper won't dissolve in diluted sulfuric acid, but a hot and concentrated sulfuric acid will do the trick. And hydrochloric acid in the absence of some other oxidants will not attack copper at all. Another factor to consider is the passivation of a metal. A thin, invisible layer of an oxide or salt can protect a metal from an acid attack. Aluminum, which is a very reactive metal, will not dissolve in oxidizing acids (e.g., nitric), because it will develop a protective layer of aluminum oxide on its surface. Non-oxidizing acids, on the other hand, will easily do the job. Metal salts that are insoluble in water can also create a passivation layer.

Regarding the original question, there are some examples for sulfuric and hydrochloric acid, where they will behave differently due to the formation of insoluble salts. One example is thallium - it dissolves in sulfuric acid but not in hydrochloric acid. An opposite is barium - it dissolves in hydrochloric acid but not in sulfuric acid.