Although this 1989 decision by the Supreme Court was in reference to buring the American flag (which, before the case, was severely limited in 48 states), it set important precedence as a decision regarding free speech in the United States. The decision was extremely controversial (it was a 5-4 court decision) because as later polls indicated, a majority of Americans were against flag burning and other examples of descrating the American flag. At issue, however, was whether or not the First Amendment of the Constitution applied to freedom to express opinions in non-verbal ways. Since 1989, there have been many attempts to introduce an amendment that prohibits flag desecration, but it has always gotten bogged down in the Senate.
As for the effect on American society -- one effect that is currently being considered by the Supreme Court again is the Westboro Baptist case. A Baptist church in Kansas has been protesting at the funerals of military members who are killed in the war. The radical beliefs of this church claim that each death of an American service person is God's punishment for America's support of homosexuality. The family of one Marine has filed a lawsuit against the church for several million dollars, which they won, but the appeals have reached all the way to the Supreme Court on the freedom of speech issue. The church members claim that they have a right to protest at the funerals under the First Amendment. They have also posted defamatory information about the Marine's family on the Internet, so this complicates the issue because free speech on the Internet is a separate concern from free speech in the form of protests. It is a vey contentious issue. The lawyer for the church is the daughter of the church's pastor. This is being considered as we speak - you might want to read about it in the newspapers.
Other ways that Texas vs Johnson has affected American society is with regard to ways of expression that are NOT verbal - such as mentioned above, the Internet.