Mr Smith was sacked for calling for a second European Union referendum, leading to a backlash from Remain MPs.
And Mr Corbyn was also facing unrest over his response to a Facebook post by street artist Mear One about the plan to paint over a mural seen as anti-Semitic.
Responding to the rows engulfing Labour, MP Wes Streeting said: “If only anti-Semites were dealt with as swiftly as Remainers.”
Asked if the Labour leader was tough on pro-EU MPs but weak on anti-Semitism, Mr Smith said: “Well he was certainly tough on me.”
Speaking to BBC Radio 4’s Today Mr Smith said: “Jeremy Corbyn clearly wants to take a different position to that which I advocate, a more Eurosceptic position if you like, on Brexit.
“My view is that the Labour Party needs to show leadership on this issue, it’s the biggest economic crisis that our country will have faced for many, many generations.”
Warning of the danger that Labour could “sleepwalk” to “effectively supporting a soft Brexit”, he added: “It’s the first instance that I can think of in living memory of a government pursuing a policy that they know is going to make our economy smaller and reduce people’s livelihoods and life chances and I cannot understand why we in Labour would support that.”
He did not endorse former Cabinet minister Lord Hain’s view that Jeremy Corbyn was conducting a “Stalinist purge”, but said he thought the leader had made a “mistake” to sack him.
Mr Corbyn had “always understood the value of people standing by their principles, it’s the position he has often adopted and it is certainly a value in him that others have extolled,” Mr Smith said.
“That’s all I have done, I have stood by principles on what I think is the most important economic and security issue facing our country for generations.”
Meanwhile, Commons Home Affairs Committee chairwoman Yvette Cooper added her voice to concerns about Mr Corbyn’s handling of the mural row.
In a public Facebook response to Mear One, Mr Corbyn had questioned why the painting was being destroyed and said the artist was “in good company” because the Rockefeller family had covered over a work featuring Lenin in their New York development.
The mural, in east London, depicted a group of businessmen playing a Monopoly-style game on a board balanced on the backs of people.
The artist denied being anti-Semitic, saying the mural is about “class and privilege” and contains a group of bankers “made up of Jewish and white Anglos”.
Mr Corbyn said he made a “general comment about the removal of public art on grounds of freedom of speech” but acknowledged he should have looked more closely at the image before posting on Facebook at the time of the row in 2012.
He said: “I sincerely regret that I did not look more closely at the image I was commenting on, the contents of which are deeply disturbing and anti-Semitic.”
Ms Cooper said she was “really troubled by the mural, the comments and the way this was handled” adding “fighting anti-Semitism is strong part of our tradition and values, and Labour must be better than this”.
Mr Smith said the mural was an “appalling, monstrous work”.