Manfred Weber criticised Britain’s apparent desire to still do business with the European Union while not being a member of the bloc.
He also claimed Foreign Secretary Boris Johnson was attempting to “provoke” Brussels with his recent choice of words while speaking in India.
However, an angry Neil railed against the MEP and asked was he not concerned about the rise of Geert Wilders and Marine Le Pen, with the Netherlands and France both facing general elections this year.
The veteran broadcaster also cited the ongoing financial difficulties in Italy as well rising criticism for German Chancellor Angela Merkel.
"Have you learned nothing from Brexit?" he asked on BBC Daily Politics.
A stammering Weber responded: "I have learned a lot, for example that nobody explained to the British people what the EU is all about."
Manfred Weber clashed with Andrew Neil on BBC Daily Politics
Have you learned nothing from Brexit
He then mocked Theresa May for extolling the virtues of free trade while simultaneously planning to take Britain out of the single market.
He continued: “For example, when we talk about trade, Theresa May made clear statements on trade, so what is EU?
“EU is the single market, that is a free trade zone – the whole ideal of the European Union, that is what the Brits always wanted.
“That was the idea of Europe – now you say you want to leave the European Union but on the other hand, you say I want to have a free trade agreement, which leads to a situation where nothing will change.
“Do you want to leave the single market or you don’t want to leave – that is the question.”
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Earlier in the interview, Mr Weber took a swipe at Mr Johnson after he compared French President Francois Hollande to someone "who wants to administer punishment beatings to anyone who chooses to escape in the manner of some WW2 movie".
The German was referring to Mr Johnson accusing President Hollande of nazi-style "punishment"
A No.10 spokesperson said the Foreign Secretary’s "theatrical" comparison "was in no way suggesting that anyone was a Nazi", pointing to the reaction to Mr Johnson’s choices of words as what Mrs May had described as a "hyped-up media report".
Discussing the comments, Mr Weber said that everyone would be better of "working together" for the benefit of the Continent.
The MEP said Britain was as much of the part of the battle against populist right-wing movements who want to destroy the Union’s "partnership approach" and accused Mr Johnson of following their combative rhetoric.
“Quite frankly, when I hear the words of Boris Johnson that is exactly what I mean,” the German said.
“To provoke each other, to say to each other that we don’t like each other – that is not the way that we should work.
Foreign Secretary Boris Johnson's best gaffes
Wed, July 20, 2016
Bumbling Boris Johnson's best gaffes and most controversial quotes.
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UK Foreign Secretary Boris Johnson said it would take him too long to apologise for the 'rich thesaurus' of rude comments and insults
“And, Boris Johnson is acting like a clown and taking away the credibility of the Government in Britain.”
Speaking on Tuesday, Mrs May said remaining in the single market would “mean not leaving the EU at all”.
“And that is why both sides in the referendum campaign made it clear that a vote to leave the EU would be a vote to leave the single market,” she added.
The German added that he had “no problem” with Brexit and is ready for a “fair deal” to be struck between the UK and Brussels.
“We are ready for a fair deal but we have to discuss what fair means,” he said.