The Pergamon Museum in Berlin displays the double herma of Socrates and Seneca (a herma, in case you're wondering, is the sculpture of a head, which makes a double herma a two-headed monster with no feet. Oh, and Socrates is the one with a beard. That's how you can tell Greeks from Romans until the end of the 1st century AD.).
There are some similarities between Socrates and Seneca, when you think about it. Beyond the obvious fact that they were both philosophers (or at least one was and the other claimed to be), they were both teachers for some rather controversial characters (Alcibiades and Nero), they both ended their lives in forced suicides, after having stepped on the wrong toes and they were both quite ugly.
But when Socrates is fizzy, influential, revolutionary for his time and challenging to this day, Seneca is... did I say he tutored Nero? I'm pretty much done talking about Seneca.
The Romans themselves had another character in mind when they thought of the “Roman Socrates”: his name was Gaius Musonius Rufus, and he lived roughly around the same time as Seneca. Very little of his work survives, which deprives us of some very exciting titles, such as “Is Marriage a Handicap for the Pursuit of Philosophy?” and “Should Every Child that is Born be Raised?”. Though, to give Rufus some credit, he also wrote “That Women Too Should Study Philosophy”, which probably caused some contemporaries to grind their teeth.
Of Rufus' most memorable deeds, we know that, in the troubled year of 69 AD, he tried to resolve the war by preaching to the troops to lay down their weapons, stop and smell the flowers, love each other with brotherly love and just chill out. The legions probably thought he was amusing rather than dangerous, so he survived.
Other than that, he spent relatively little time in Rome: he chose a self-imposed exile to protest against Nero's treatment of Plautus; then he was actually exiled by Nero for an alleged involvement in a conspiracy, returned to make his memorable contribution to the war, only to be banished again when Vespasian decided to kick all philosophers out of Rome.
Vespasian's decision troubled no one at the time, the consensus was that philosophers were a waste of time and space, breathing in the air of the responsible citizens. Not evil enough to be killed, but not worth keeping in Rome, either.
In describing the life of his father-in-law, Agricola, Tacitus quickly mentions this:
“I remember he was wont to declare, that in his early youth he studied Philosophy with more avidity than was allowable to a Roman and a Senator; till the discretion of his mother checked his spirit”. Well, good for Agricola's mother! I guess she wasn't one of the women that took Rufus' advice and studied philosophy.
Quintilian makes it even clearer:
“Which of the philosophers, indeed, ever frequented courts of justice or distinguished himself in public assemblies? Which of them ever engaged even in the management of political affairs, on which most of them have given such earnest precepts? But I should desire the orator, whom I am trying to form, to be a kind of Roman wise man who may prove himself a true statesman, not by discussions in retirement, but by personal experience and exertions in public life.”
There you go. Roman wise men were not philosophers. One or the other might have been occasionally compared to Socrates, but I'm not sure this was a compliment.
As for Gaius Musonius Rufus, he wasn't completely forgotten after all his efforts, I can see he's still extensively discussed on the site of the International Vegetarian Union.