Although exclusive breast feeding for 6 months is recommended , Apart from reduction of Obesity, I would like to know what health benefits does extended breastfeeding duration (2yr and beyond) as recommended by WHO provide to both the mother and the baby? I would also like to know some personal experiences of parents who have done this.
An important thing which most analyses of breastfeeding studies that I have read note is that women who breastfeed tend to have higher IQs and incomes than women who don't, at least in first world countries. Because of the changes in lifestyle factors which result, it is nearly impossible to sort out what impacts on women and children are from breastfeeding and what are from better access to quality foods, better access to educational resources, and so on. This is even more true of women who breastfeed for an extended period of time.
From the World Health Organization comes a paper on the long-term effects of breastfeeding - in general, with some studies only going to 3 months - dated 2013. The Skeptical OB has a summary. In the case of most health benefits that are commonly claimed to be associated with breastfeeding and even more with extended breastfeeding the WHO determined that confounding variables such as maternal intelligence, education, and income could not be ruled out. These benefits included a slight reduction in obesity, a tiny increase in IQ, and a small decrease in blood pressure. The conclusion is:
The most recent Harvard study referenced in the other answer also has serious flaws as reported by the Skeptical OB. As with other breastfeeding studies, the major flaw is that this study did not take out the confounding variables.
After doing some research I have found two studies which attempted to separate out socio-economic status and educational status in the mother from breastfeeding status. Unfortunately neither measured breastfeeding duration over 2 years and both are based on the same group of children. The studies looked at a less than six month duration and a 12-18 month duration of breastfeeding. Both studies  and  did show the minor IQ bump, as mentioned above in the quote from the WHO paper.
I am not convinced that all the benefits of breastmilk have been discovered yet, as it seems that something that is newly discovered and amazing about breastmilk is being discovered every year. However so far the scientific evidence does not support a significant public health benefit over the long term to extended breastfeeding.
The initial challenges to breastfeeding tend to be in the early months - getting the milk supply established, getting past cluster feeding and growth spurts, dealing with pain for the mother, the transition to going back to work, and so on. In my opinion overstating the small benefits of breastfeeding in general and extended breastfeeding in particular will not help families who desire to breastfeed be any more motivated to overcome these challenges.
No matter what the studies say, for some families there will be benefits to breastfeeding up to and past age two. There are all sorts of intangibles, many mentioned here that are backed up by experience, not science. Breastfeeding provides cuddle time with notoriously on-the-go toddlers. It may help with hydration during illness. It can be a tool for calm in the emotionally stormy times of toddlerhood. If extended breastfeeding is working, that's great. There's no need to exagerate the health benefits to defend it.
Breastfeeding up to two years of age or beyond is based on the fact that breast milk is a key organic source of DHA (Omega fatty acids), vitamin A, calcium and proteins and provides protection against infectious agents.
In 2008 the American Academy of Family Physicians said this in their position paper:
Many scientific studies prove the point that longer breastfeeding enhances the neurodevelopment of infants and their intellectual and scholastic ability in later life. from one of these studies increase in the IQ by breastfeeding is directly related to the duration of breastfeeding. These benefits are strongest for the infants of low birth weight on breastmilk feeding. It is found that Poor who are more likely to enter into child malnutrition are more likely to benefit from enhanced optimal breastfeeding practices.
longer breastfeeding promotes suppression of following Diseases and benefits to the infant :
WHO recommends that breastfeeding should continue for two years or more without accurately establishing a maximum duration for breastfeeding.In addition, there is insufficient evidence on the advantages and disadvantages of continuing breastfeeding after two years.
It is unfortunate that about many mothers are unable to breast feed even upto 1 year. This may be because We’ve sexualized the breast so much that people have forgotten breasts are for breast-feeding. There is A Harvard Study which looks at both health AND cost outcomes for U.S. mothers if they were to breastfeed for 1 year.The study found that if 90% of mothers breastfed, U.S. women would significantly lower their risk of heart attack, breast cancer, and hypertension. On a macro level, further investment in lactation support for moms could prevent $17.4 billion in maternal health costs.
Many mothers have shared positive Experiences of extended breastfeeding in this article.
Note:I am updating as an answer what I found on the internet.