[After performing several edits to the original version of this post based on more recent statistics and survey results, and noting the edits in brackets, it started to get confusing. So I am simply duplicating the prior post and changing the data as needed, without noting the changes. Those who are interested in the earlier numbers can refer to the original post.]
I decided to see what I could find out, specifically, about comparative Mormon and Catholic divorce rates. I found various sources with varying conclusions. The sources also calculate the “divorce rate” in different ways. I’m not concerning myself with which way is best (nor yet even understanding all of them) since all I’m interested in are the comparisons.
Here is what I have found:
I had heard repeatedly that Utah, which of course is predominantly Mormon, had the lowest divorce rate of any state. However this table from the Centers for Disease Control indicates that Utah’s divorce rate for 2012 was 3.3 (per 1,000 people), which is equal to or higher than 17 other states and the District of Columbia. 
This article from the Encyclopedia of Mormonism on the Brigham Young University website, says the following: “Recent U.S. data from the National Survey of Families and Households indicate that about 26 percent of both Latter-day Saints and non-LDS have experienced a divorce (Heaton et al., Table 2).”
This article from the FAIRMormon Blog discusses a survey of Mormons published in 1985 (using, if I understand correctly, data from 1981), which, when compared with the results of previous surveys of non-Mormons, showed that Mormons had a lower divorce rate than Catholics or Protestants, at 14.3% for men and 18.8% for women; compared with 19.8% and 23.1% for Catholics. 
The FAIRMormon article also includes the results of “the 1999 Barna Survey”, which shows Mormons having a divorce rate of 24%, compared with 21% for Catholics.
This blog post includes a graphic of a “2008 Religious Landscape Survey” (“Sources: Pew Research Forum, Barna Research Group”). The graphic indicates that it was published in The Denver Post. This survey also shows Mormons with a divorce rate of 24%, compared with 21% for Catholics.
The webpage of the Religious Landscape Survey itself doesn’t give divorce rates, but says that 12% of Catholics had a marital status of “divorced or separated”, compared with 7% of Mormons. 
This doesn’t include a comparison with Mormons, but I’m including it since it’s pertinent to the general theme: “The Georgetown center reported in late September that a variety of national surveys show ‘Catholics stand out with only 28% of the ever-married having divorced at some point.’ While 28% remains a troubling statistic, the research suggests that this figure compares favorably with the 40% divorce rate for those with no religious affiliation, 39% for Protestants and 35% for those of other religious faiths.” “Divorce Statistics Indicate Catholic Couples Are Less Likely to Break Up“, National Catholic Register (ncregister.com), November 14, 2013. See also, this.
In all, there doesn’t seem to be a huge difference between overall Catholic and Mormon divorce rates.
In several places (including the FAIRMormon article linked to above) I saw a cited divorce rate of 6% for Mormons who are married in the Temple. Some claim that this figure is misleading since it is based on Church records, which would only show Church divorces, ignoring the possibility that people married in the Temple might simply get a civil divorce, which would not be accounted for in Church records. However I found one article which asserted that the figure did take account of civil divorces. I’m not citing any sources for this either way, because I haven’t found any reference to specific data that it is supposed to be based on. (If anyone has information showing who is right, please share it in the comments.)
Of course the Catholic Church has no equivalent of Temple marriage, by which to directly compare the divorce rates of exceptionally devout Catholics with their Mormon counterparts. There is, however, this survey of 505 Catholic women aged 21-66 who had been instructed in the use of Natural Family Planning (NFP). The survey found that 99.8% had never been divorced; reduced to 97% when counting (for purposes of comparison with data concerning non-NFP-using Catholics) only those aged 21-44. Ninety-nine percent of participants aged 21-44 were married at the time of the survey; the remaining 1% were widowed. 
This can’t be compared directly with Mormons married in the Temple. For one thing, Catholics who use NFP are probably a more “elite” group, or in other words a smaller subset, than Temple-married Mormons. Still I think it shows something of what results when Catholics take their faith as seriously as Temple-married Mormons.
One issue regarding all of these statistics, is how to identify who counts as a “Catholic” and who as a “Mormon”. In this regard the Religious Landscape Survey says the following: “In this survey, we rely on respondents’ self-reported religious identity as the measure of religious affiliation. Catholics, for instance, are defined as all respondents who said they are Catholic, regardless of their specific beliefs and whether or not they attend Mass regularly.” I suspect most random sample surveys do the same.
So the question becomes, whether Catholics or Mormons are more likely to continue identifying themselves as Catholic or Mormon once they have stopped “practicing”. If there is a much larger proportion of people continuing to call themselves Catholic while disregarding Church teaching and hardly ever going to Mass — as I suspect there is — that might skew the Catholic numbers more than the corresponding number of inactive yet self-identifying Mormons skew the Mormon numbers. The 2014 Religious Landscape Study seems to bear this out. It finds that 84% of those who self-identify as Mormons consider their religion “very important”, whereas only 58% of self-identified Catholics say the same.
 Divorce rates by State: 1990, 1995, and 1999-2012; source: CDC/NCHS, National Vital Statistics System.
 For what it’s worth, the Mormon and non-Mormon data come from different sources, the Mormons having been contacted through Church records; the Mormon responses were then compared with non-Mormon data from prior studies done by different researchers.
 Pew Forum’s 2014 U.S. Religious Landscape Survey, which “is based on telephone interviews with more than 35,000 Americans from all 50 states. This is the second time the Pew Research Center has conducted a Religious Landscape Study. The first was conducted in 2007, also with a telephone survey of more than 35,000 Americans.”
 “Divorce Rate Comparisons Between Couples Using Natural Family Planning & Artificial Birth Control”, by Mercedes Arzú Wilson, Family of the Americas Foundation, Dunkirk, MD (2001).