Prime Minister David Cameron has said a proposed change in the UK voting system would be "bad for democracy" - as he lines up against his own deputy.
An hour earlier, Nick Clegg called for the first-past-the-post system to be replaced by the Alternative Vote method, which he said was "fairer".
Both men insisted their opposing views would not wreck the coalition - whichever way the public vote.
The referendum on changing the voting system will be held on 5 May.
If the public votes for change it will mean an end to Britain's traditional "first-past-the-post" voting system, which would be replaced by the Alternative Vote (AV).
AV is not a form of proportional representation, as the Lib Dems have traditionally demanded, but a preferential system, in which voters rank their choices.
Those who want change claim it will mean an end to wasted votes and ensure that all MPs have the backing of at least 50% of their constituents.
But opponents say it will produce freak and unrepresentative results and lead to more hung Parliaments.
Mr Cameron and Mr Clegg were arguing their respective cases a day after Parliament approved legislation paving the way for the referendum to be held on the same day as devolved elections in Scotland, Wales and Northern Ireland and local elections in parts of England.
In his speech in London, Mr Cameron said the coalition had worked well together but there was a "real difference of opinion between us" on this issue.
He said he believed the Alternative Vote was "completely the wrong reform" and would be "bad for our democracy" - leading to unfair results and an unaccountable political system.
Under AV some votes counted for more than others, he said - as those who voted for less popular parties would see their second, third or fourth preferences counting towards the result.
"If you vote for a mainstream candidate who is top of the ballot in the first round, your other preferences will never be counted," he said.
"I don't see why voters of the BNP or Monster Raving Loony Party should get their votes counted more times than supporters of the Conservatives or for that matter, Labour or the Liberal Democrats."
"The principle of one person, one vote is what makes our democracy fair. AV flies in the face of that."
"Everyone gets" first-past-the-post, Mr Cameron said, while AV was confusing to explain and was only used by three countries - Australia, Fiji and Papua New Guinea, while more than 60 countries used first-past-the-post.
He argued it would not end "safe seats" for MPs, as in Australia, nearly half of all seats were considered "safe" and smaller parties had been "all but obliterated".
Research suggested AV would have produced larger Labour landslides between 1997 and 2005, and larger Conservative ones in the 1980s and could lead to "even more disproportional" results.
And he argued, the way votes are counted could mean "those who are courageous and brave and may not believe in or say things that everyone agrees with are pushed out of politics and those who are boring and the least controversial limping to victory".
"It could mean a Parliament of second choices."
In his speech he also suggested that if AV was used, "we may have to buy and install electronic voting machines to make sense of all the different outcomes and possibilities".
But when the topic was raised in a question session after Mr Clegg's speech, the deputy PM dismissed it. He said in Australia, which uses AV, votes were counted by hand. Reports it would cost millions to administer AV were "wildly inaccurate", Mr Clegg said, adding he hoped the No campaign would not "create a whole barrage of scare stories and myths about this".
In his speech in Leeds, Mr Clegg claimed the current voting system created "jobs for life" - claiming most of the MPs caught up in the expenses scandal had been in safe seats.
"When a system makes corruption more likely, it should be changed," said the deputy PM.
He said elections boiled down to parties trying to get their own vote out in marginal seats - leaving vast swathes of voters ignored and constituents being "taken for granted" and many voters had "given up caring".
A switch to AV would mean politicians had to work hard to appeal beyond their core supporters and it would put an end to tactical voting, he said.
He said the referendum was a "once in a generation" change comparable to the emancipation of women adding: "First-past-the-post was perfect for a time when the choice was only ever between two parties, but that hasn't been the case for a long time."
He dismissed the argument that it would lead to more hung parliaments: "Australia has had AV for 80 years and they have had fewer hung parliaments than we have had with first-past-the-post."
Questioned after the speech, he said AV make politics "a less tribal, a little less partisan, a little more open minded".
Mr Clegg, whose party backs the "single transferable vote" system, described AV as a "miserable little compromise" during last year's general election campaign, when Labour announced it would hold a referendum on AV.
He said that had been a specific response to a question about a "very last minute" suggestion from Labour, after 13 years in power, which everyone knew was an "empty gesture". He argued AV was "a small change which will make a big difference".
Although both sides have been preparing for the poll for months, the No camp launched its referendum campaign earlier this week while the Yes camp is due to do so next month.
Labour leader Ed Miliband has said he will campaign for a switch to AV. However, his party is split on the issue with its MPs lining up on both sides of the argument.
Nigel Farage, leader of the UK Independence Party, said he was not particularly enthusiastic about AV - but would be campaigning for a "yes" vote as it was "a stepping stone towards full proportional representation".
Under first-past-the-post, voters select one candidate and the individual with the most votes wins.
Under the AV system, voters rank candidates in their constituency in order of preference. Anyone getting more than 50% of first-preference votes is elected.
If no-one gets 50% of votes, the candidate with the fewest votes is eliminated and their backers' second choices allocated to those remaining.
This process continues until one candidate has at least 50% of all votes in that round.