Why are scientists interested in identical twins?

In the case of identical twins, the classic research involves psychology and the standing question: nature vs. nurture (whether genetics or the social environment plays the greater role in cognitive development). Identical twins are genetically perfect, and therefore can be observed to see how they develop. Sometimes, twins are separated at birth because of adoption. The two are therefore adopted by two different families (different social environments). When researchers stumble upon such a pair, it turns out that most of their behavior (ex. habits, sexuality, etc.) are similar, suggesting that genetics plays a significant role in social development.

Twin studies versus sibling studies explores genetic versus environmental influences.
One way to isolate genetic factors is to try to control for the environmental factors by studying twins and same age siblings to see what degree of the character or behavior is genetic versus environmental.

 It is important to compare both dizygotic DZ twins with monozygotic MZ twins ,when possible, since they both experience a twinned environment during embryogenesis and after birth. Comparing MZ to DZ to siblings allows the genetic differences to stand out from environmental influences.

Twin studies of autoimmune diseases indicate there are environmental triggers that influence whether the genetic condition .
"Twin Study Spotlights Alzheimer's Genetic Link'
http://pn.psychiatryonline.org/content/41/7/32.full

MS twin study shows environment plays key role
http://www.bionews.org.uk/page_59451.asp

"A twin study of cognitive function in chronic fatigue syndrome: the effects of sudden illness onset."
http://www.ncbi.nlm.nih.gov/pubmed/17605583
http://www.bionews.org.uk/page_59451.asp

Even scientists like some of the Olsen Twins movies.
I used to like watching the "Sweet Valley High" TV show.

I'm sure someone else is going to list:
a) they're genetically identical, and so are useful in so-called "nature vs nurture" studies, blah blah blah.
b) they're useful in studies of epigenetic changes occurring over a lifetime (identical twins get less identical as they age)