I'll recommend a few books to you: Nora Titone's My Thoughts Be Bloody: The Bitter Rivalry That Led to the Assassination of Abraham Lincoln, James Giblin's Good Brother, Bad Brother: The Story of Edwin Booth and John Wilkes Booth, and Gene Smith's American Gothic: The Story of America's Legendary Theatrical Family ─ Junius, Edwin and John Wilkes Booth. You might also consider Joseph Donohue's The Theatrical Manager in Britain and America: Player of a Perilous Game. I only mention this because it has a chapter devoted to Edwin Booth's career and the author of the book is a cousin twice-removed.

The latter book explains that after the death of his wife, Edwin stayed close to New York City, where his daughter Edwina was being cared for by his mother and sister Rosalie. Edwin was a staunch Unionist and one of (if not the) most famous actors in New York City.

"In the Jersey City railroad station he once saw a young man buying a sleeping car berth from a conductor on the platform. There was a crowd jamming in, and then the train began to move. The young man was twisted off his feet. He dropped down between the platform and the car, helpless and in immediate danger of being crushed under a rolling wheel.

Edwin leaped forward, grabbed his coat collar, and pulled him up to safety on the platform. The young man looked up, recognized his rescuer at once, and said, 'That was a close call, Mr. Booth.'

Edwin always hated being recognized. He went on his way. Weeks later he heard from Adam Badeau that the young man had told the story around Washington and that word had drifted down to Grant's headquarters. It was Robert Todd Lincoln."

That account comes from a story in the November 1893 issue of Century Magazine, whose page 133 includes a continued story entitled "Memories and Letters of Edwin Booth".

In 1909, Robert Lincoln provided his own account of the rescue to one Richard Watson Gilder. This letter was reprinted in the March 1948 issue of the Journal of the Illinois State Historical Society.

"The incident occurred while a group of passengers were late at night purchasing their sleeping car places from the conductor who stood on the station platform at the entrance of the car. The platform was about the height of the car floor, and there was of course a narrow space between the platform and the car body. There was some crowding, and I happened to be pressed by it against the car body while waiting my turn. In this situation the train began to move, and by the motion I was twisted off my feet, and had dropped somewhat, with feet downward, into the open space, and was personally helpless, when my coat collar was vigorously seized and I was quickly pulled up and out to a secure footing on the platform. Upon turning to thank my rescuer I saw it was Edwin Booth, whose face was of course well known to me, and I expressed my gratitude to him, and in doing so, called him by name."

Because both families were celebrities (one from politics, one from entertainment), the Booths and Lincolns did have some encounters beyond the infamous Ford's Theater assassination. Kevin Morrow wrote about these in an article for the New York Times in 2013, which you may find interesting.

After the Lincoln assassination, Edwin suspended his acting career for a time. Following the interregnum, he resumed acting and performed on stage across the United States and in Europe. He endured some heckling ─ sometimes fierce ─ from audience members who believed he was in some way connected to the assassination, but he endured nonetheless, even as he harbored deep guilt and grief. He never played in Washington, D.C. again. Even when President Chester A. Arthur asked Booth to return to D.C. near the end of his acting career, Booth refused. He played in Baltimore instead, and Arthur traveled to see him.

Booth continued his acting career until 1888. He died in 1893.

Answering the second part of your question, whether the two men exchanged words again, is much more difficult because you cannot prove a negative. I have run across no accounts of them exchanging words, however.

Robert Lincoln had a fascinating series of coincidences to his name as well. He had been invited to Ford's Theater on the night of the assassination but begged off (Gen. Grant and his wife also declined the invitation), which is part of the reason why Abraham and Mary were accompanied by Maj. Henry Rathbone and Clara Harris.

Robert Lincoln was Secretary of War under James Garfield, and was in Garfield's train when the president was shot by Charles Guiteau. Garfield died two months later, from complications. In 1901, Lincoln was invited to attend the Pan-American Exposition in Buffalo, New York, with President William McKinley. At that event, McKinley was shot and killed by Leon Czolgosz.

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P.S. /u/sonnyclips posted a neat bit of trivia in a comment deleted by another moderator as being insufficient to stand on its own. Apparently, Junius Booth ─ father of Edwin and John, among others ─ once wrote a letter to President Andrew Jackson promising to shoot him:

"You damn’d old Scoundrel if you don’t sign the pardons of your fellow men now under sentence of Death - De Ruiz & De Soto - I will cut your throat whilst you are sleeping. I wrote to you repeated Cautions so look out or damn you. I’ll have you burnt at the Stake in the City of Washington."