In American politics, the obvious assumptions are often also the most lazy and wrong-headed. Look at Rick Perry, for example. It is easy for many on the left, especially outside America, to see the resurrected spectre of cowboy President George W Bush. Both have the familiar Texan swagger and a rustic ranch twang in their accents. Each places a common sense folksy charm at the core of his political persona. Both put their faith front and centre. Both used the Texas governorship as a launch pad for a bid for the White House. They seem cut from the same political cloth: a rightwing, simplistic, religious-infused worldview – out of place with America's diversity and complex challenges.

But this view is wrong. Perry is actually not much like Bush at all. Liberals often forget that much of Bush's Texan image was an affectation. Bush was born in Connecticut, the son of one of America's most elite families. He went to a boarding school in Massachusetts and then Yale, where he joined the Skull and Bones secret society, which also boasts "Boston Brahmin" John Kerry as a member. Bush's elite status and Yankee family assured him opportunities (such as managing a top baseball team) that others only dream of.

Perry is not like that. Perry was born in a West Texas town, Paint Creek, so tiny that it literally did not appear on many maps. His family were hardscrabble farmers and Perry grew up tough. He did not go to Yale (the embodiment of New England elite). He went to Texas A&M, a hardcore college of Texas values and tough discipline. Perry once, famously, shot a coyote while jogging near Austin and did it with his own gun. It's tempting to imagine that in a similar situation, Bush would have had somebody else shoot it for him. Bush's tough guy, cowboy image was deliberate myth-making. Perry is the real deal.

Then, there is the issue of religion. Bush came late to his active Christianity, turning born again after his 40th birthday. Though, by European standards, Bush was openly religious in his public life, he is no Perry. Perry, after all, happily declared three days of prayer to try and break a crippling drought (surprise: it didn't work). While Bush could perhaps privately pray for rain, it is hard to see him making it official public policy. Perry also did not shy away from addressing "The Response", a religious rally at which he called on Jesus to guide the nation and asked God's forgiveness for the sins of the American government. Perry has happily merged church and state – using his office to promote the rally. While Bush frequently prayed with top aides, it was behind closed doors.

There is also the issue of political style. I was in Texas last week to look at some of the ugly economic realities that lie beneath Perry's claims for a job-creating "Texas miracle". While there, the Perry-Bush comparison issue frequently came up. What was striking was that there was an acceptance – among people with no fondness for either of them – that Bush as Texas governor was genuinely open to reaching out to Democrats. He sought workable relationships with his opponents. He always ran as someone who wanted to unite and not divide (even if he did not achieve that in office).

Perry has not done that. His career is a long trek to the right from being a conservative Democrat to a Tea Party darling. In office, he has concentrated power in his own hands, appointing close allies to top posts and skipping the need for bipartisanship to get policies through. On policy matters, just look at education. Bush (whose wife was a teacher) used education as a centre-piece of his time in office in Texas. You can argue the policy merits of his decisions, but Bush tried to reform a system.

Perry's contribution to Texas education? Lopping $4bn off the education budget.

But perhaps the killer evidence that Perry and Bush are not alike is that the two camps do not like each other. Both Bush and Perry were mentored by Karl Rove. But Rove has publicly criticised Perry and backed Kay Bailey Hutchison in a (failed) bid to challenge Perry in the governors' primary race of 2010. Indeed, many of the Bush circle lent Hutchison their support in a rather obvious statement of distaste for Perry's ambitions.

And the animosity is not one-way. Perry takes swipes at Bush's record, especially on spending. Observers of Texas politics say that when Perry refers to Bush in a speech as "George", then it's a dig at his predecessor as the Bush family, apparently, cannot stand it when Perry calls him by his first name in public. Bush even said last week he would not be endorsing anyone in the 2012 Republican nomination race, Perry included.

In short, Bush and Perry are far from ideological brothers. Instead, Perry is the sort of politician Bush pretended to be. With President Barack Obama's poll ratings nose-diving, just as Perry's rise, that is not a very comforting thought.