Redheads do experience certain sensory stimuli differently. It's not as simple as "they feel less/more pain", but it's rather dependant on the nature of the stimulus. But first, let me begin by explaining a bit more about how all this red hair thing is even possible.
The red colour of these people's hair is the result of a mutation in a gene located in chromosome 16, called Melanocortin 1 receptor (MC1R) or Melanocyte-stimulating hormone receptor (MSHR). The name of this gene is the same one that's used for the protein that it encodes. This G protein-coupled receptor is embedded in the cell's membrane and can bind, at the outer side of the membrane, to a class of pituitary hormones known as melanocortins, most importantly the Melanocyte-stimulating hormone (MSH). This receptor is very important in the pigment-producing cells on the skin, hair follicles and eyes, known as melanocytes, as it takes the signal it receives from the hormone on the outer surface of the cell membrane and transmits it through the membrane and into the cell. Of course, this is simplified; I don't want to bore everyone, especially those people who are already familiar with these mechanisms (I know there are many on Quora). The thing here is that this protein (receptor) is responsible for making melanocytes produce melanin, the blackish pigment that gives skin its colour. When a person gets their two copies of the MC1R gene mutated (recessive inheritance), the whole system just can't work the usual way. Note that both copies have to be mutated, the one that comes from the mother and the one from the father. If this happens, the melanocytes don't produce the same quantity of the darker eumelanin pigment as the ones in normal people. Instead, more of the lighter phaeomelanin is produced. This is what confers a lighter colour on the hair. Because of this, light-skinned people can produce vitamin D more easily (they need less sunshine), but are also less tolerant of UV rays and more prone to developing melanomas. It doesn't end there, though, as the MC1R appears also to have a role in non-pigmented cells. It's been proposed that it interacts with endorphins (endogenous morphins, AKA natural painkillers), which could explain why redheads experience pain differently. This last part hasn't really been elucidated, thus far.
There are many rumours about how redheads feel pain differently -some of them going back a very long time-, but only very few things have been actually proven by respectable research. One of the studies, funded by the National Institutes of Health (USA) and published in 2004[1], found that "Anesthetic Requirement is Increased in Redheads": they compared 10 redheads to 10 brunettes, transmitting painful electrical impulses through needles on the skin. They found that redheads needed more of the anaesthetic desflurane to stop feeling pain. This graphic says it all:
Yet another NIH-funded study, published the following year[2], came across a few other interesting findings: they compared 30 gingers to 30 brunettes and found that redheads are overall more sensitive to cold pain and to heat pain, meaning that redheads started experiencing pain at a lower temperature than brunettes when a square thermode was placed on their skin and heated up, but they experienced pain at a higher temperature than brunettes when they started diminishing the termperature on that same thermode. This tells us that redheads have a narrower comfort zone when it comes to room temperature, and will complain about being cold more easily.
A third study sheds even more light on the subject. Danish researchers published in 2010[3] that redheads felt less pain compared to blondes/brunettes, but this phenomenon was observed in quite an unusual situation: they applied irritating capsaicin cream on the skin and punctured the painful area of the skin with a needle. Redheads, in this case, felt less pain than the other women. The authors suggest a role of MC1R in the brain that would regulate these experiences, but again this isn't conclusive, at least for now.
It's worth noting that in all three studies they used women, not men, so technically it hasn't been proven that red-haired men experience these stimuli differently. All in all, it's reasonably sensible to assume so.
Now go, explain your ginger friend how she's different from a brunette!
References
[1] http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1362956
[2] http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1692342
[3] http://www.scandinavianjournalpain.com/article/S1877-8860(10)00111-4/