Cameron took a swipe at the Labour leader's political style by slamming those who "shout into megaphones, wave banners and sign petitions".
In his message welcoming 2016, he criticised those who are "on protest march" and claimed that only the Tories are "the ones who are able to make the arguments and take the difficult decisions".
While Cameron spoke of Britain's "renewed strength" and the "difficult" negotiations with Europe in his New Year message, Corbyn reflected on 2015 as a year when Britons were continually "helping out" others.
"Happy New Year everyone, hope it's a good one," the Labour leader said, before praising those who have assisted others with problems like the recent flooding in the north of England.
Corbyn said Christmas has been a time of "great spirit, families coming together, communities supporting each other" and highlighted those "helping out neighbours, helping out the homeless, helping out those who are victims of floods in so many parts of the country".
He added that 2016 would be "the start of a journey to deliver a Labour government in 2020 - a Labour government that will deliver a fairer, more just, more prosperous society that we can all enjoy. A society that works for all, not just the few".
"I was elected leader of the Labour Party three months ago on a mandate for change and during this last three months we have challenged the Government on working tax credits and defeated them, we challenged them on cuts to the police service and defeated them, we challenged them on running the prison service in Saudi Arabia and we defeated them," Corbyn said.
"We have to challenge them much more next year - much more on their cuts to local government and their lack of investment in the needs of our economy and our people. We want to build an economy fit for the 21st century."
Meanwhile David Cameron's message said 2016 would be a "game changer" for Britain which will begin the year with "renewed strength."
He laid out his priorities for the next 12 months as "delivering the education, training, jobs, tax cuts, healthcare and housing people need".
"But we're also going to make sure no one should be on the outside looking in at all these things – that everyone is a part of Britain's rise. In doing so, we can make 2016 a game changer for our country."
He also seemed to take a swipe at Corbyn, criticising those who "shout into megaphones" and "sign petitions" and insisting that while "others are on protest march, we remain on the long walk to a Greater Britain".
"There are many people who will tell you how deeply they care about these issues. They will shout into megaphones, wave banners and sign petitions.
"But we're the ones who are able to make the arguments and take the difficult decisions in order to defeat these social scourges and deliver real security. So while others are on protest march, we remain on the long walk to a Greater Britain. We won't get there overnight. But during 2016, we will make some of our most significant strides yet."
Cameron also touched on his hope to secure a package of reforms in the European Union when he meets fellow leaders in February.
He said: "We're fighting hard to fix the aspects of our EU membership that cause so much frustration in Britain – so we get a better deal for our country and secure our future. It is a difficult negotiation with 27 other countries. But throughout we are driven by one consideration: what is best for Britain's economic and national security."
In December, Corbyn gave The Huffington Post UK an alternative New Year message, saying: "there's a lot wrong with this planet, there's a lot wrong within this country" but that he feels his job is to unite "a whole wave of people... who want to see a better and different society".
The message was part of an in-depth interview run in three parts, to mark 100 days since the Labour leader was elected.
Jeremy Corbyn Interview (Part 1):On His First 100 Days, Leadership, Donald Trump, Bernie Sanders, Hillary Clinton And Tyson Fury
Jeremy Corbyn Interview (Part 2):On War, George Galloway, Trident, MP Reselections And More Of A Say For Labour Party Members
Jeremy Corbyn Interview (Part 3): On The Housing Crisis, Media Plurality, Climate Change, Religion, Bolivia and 'Corbynistas'
Liberal Democrat leader Tim Farron used his message to argue "Britain deserves better" as he criticised the Tories and Labour for being divided parties.
Farron insisted his party, which was all but wiped out in Westminster at the general election, would enter 2016 with a "a new sense of purpose, a new drive and a sense of ambition".
He criticised government cuts to Universal Credit and policing and said more should be done to support refugees fleeing Syria.
Farron said: "Could there be a more perfect image of the need to challenge the way the Tories are running Britain than seeing David Cameron happily enjoying a Christmas Party with Rupert Murdoch?
"Out of coalition, the Tories are arrogant, deceitful and out of touch.
"Britain deserves better."