Question

though the outcomes of schenck and new york times differed, what did these decisions have in common?
the government has a heavy burden to prove harm.
the government can limit speech that causes harm.
the government has unlimited power to limit speech.
the government must follow the first amendment.

Explanation:

In Schenck v. United States (1919), the Supreme Court established the "clear and present danger" test. It allowed the government to limit speech that presented a clear and present danger, like Schenck's anti - draft leaflets during WWI. In New York Times Co. v. United States (1971), the Court protected the New York Times' right to publish the Pentagon Papers. The commonality is that the government can limit speech that causes harm (as in Schenck when there was a danger to the war effort, and in a general sense, the principle that harmful speech can be restricted). The "clear and present danger" in Schenck and the balancing of interests (in New York Times where the public's right to know was weighed against national security, but the idea that speech with harmful consequences - like revealing overly sensitive info that could harm troops - could potentially be restricted in some cases, though in New York Times the govt failed to prove such harm) shows the government can limit harmful speech.
The "heavy burden to prove harm" was more of a later development in free - speech jurisprudence (not the commonality in these two cases). The government does not have "unlimited power" (both cases are about some form of speech protection or limitation based on harm, not unlimited govt power). While the government must follow the First Amendment in general, the more specific commonality between these two cases is about speech limitation based on harm.

Answer:

The government can limit speech that causes harm.