Do some U.S. states' constitutions interpret the U.S. Constitution's First Amendment differently? If so, what do they say?
A US state's constitution cannot "put restrictions on" the Federal constitution, or any of the rights guaranteed by it, if by that is meant limiting the rights Federally guaranteed. The so called "Supremacy clause" of Article VI says:
This Constitution, and the laws of the United States which shall be made in pursuance thereof; and all treaties made, or which shall be made, under the authority of the United States, shall be the supreme law of the land; and the judges in every state shall be bound thereby, anything in the Constitution or laws of any State to the contrary notwithstanding.
(The above was more relevant before the question was edited to remove the 'put restrictions on" language. But it is still accurate and somewhat relevant.)
Similarly, and because of the Supremacy clause, the content of a US State's constitution cannot change the interpretation of the federal constitution, in either state or federal courts, including the presence or absence of rights similar to those protected by the Federal Constitution.
However, a number of state constitutions have guarantees which parallel the federal First Amendment, particularly its freedom of speech and freedom of religion aspects. This is particularly true of state constitutions written prior to the ratification of the Fourteenth Amendment. At that time, the Federal Bill of Rights only restricted the federal government, and so when similar restrictions were wanted against a state government, they had to be in the state constitutions. See Barron vs Baltimore for the limitation of the bill of rights to restricting the federal government at this time.
Such provisions did not usually reference the Federal First Amendment explicitly, they simply had more or less similar language.
Since the passage of the 14th, and the incorporation of the various provisions of the Federal bill of rights against the states (largely starting in the 1920s) most court cases have refereed to the Federal provision rather than any similar state provision, unless, as is sometimes the case, the state provision offers greater rights or wider scope than the federal one does.