Why is it advised that infants are fed mother's milk?
The phrase "Breast is best" is a hotly debated one (source: The Guardian and personal communications with many folks). The reason why we don't want to feed infants cow's milk is, however, anything but debated, because  

Cow's milk does not provide enough:


Vitamin E
Iron
Essential fatty acids


And because it contains too much:


Protein
Sodium
Potassium (source: NIH's MedLine)
Further, mother's milk is generally free of pathogens, cow's milk may not be (source: Australian Unity). However note that, unfortunately, HIV can hitchhike within the maternal white blood cells to the infant (Quintanilla, 1996)


There are many benefits to mother and infant of breast milk. These benefits are, however, typically compared to the benefits over formula. No study will compare the benefits of mother's milk over cow's milk, simply because infants should never be fed cow's milk. 

The benefits of breast milk for the infant are the following:  


Breast milk a nearly perfect mix of vitamins, protein, and fat, all provided in an easily digestible form, as compared to formula; 
Breastfeeding lowers baby's risk of having asthma or allergies; 
Plus, babies who are breastfed exclusively for the first 6 months, without any formula, have fewer ear infections, respiratory illnesses, and bouts of diarrhea. They also have fewer hospitalizations and trips to the doctor (source: WebMD).
Breast milk contains a variety of growth factors, including EGF, BDNF and GDNF that may promote vascularization and neural development (Ballard &amp; Morrow, 2014).


There are a host of immunoprotective factors present in breast milk. These proteins offer protection against diarrhoea, food allergies and infections. The immunoprotective components of human milk include, but are not limited to:


Lactoferrin : binds to iron, thus rendering it unavailable to viruses, fungi and bacteria, which is an effective means of inhibiting their growth (Ballard &amp; Morrow, 2014);
Lysozymes: destroy viruses and bacteria by disrupting their integrity (source: NIH's MedLine);
Secretory IgA: immunoglobulins that destroy viruses and bacteria. Secretory IgA is believed to survive the harsh conditions of the intestines, especially in infants and is therefore active in the intestines of the baby. IgA can bind to pathological viruses and block their receptors necessary to invade the epithelial cells lining the intestinal wall. It also clumps bacteria together through cross-linking, which may affect their dispersion. Bacterial cell membranes, like shown in Salmonella, may be disrupted by IgA binding (Mantis et al., 2011). As mentioned by anongoodnurse in the comments; because mother and baby share the same habitat and hence encounter the same pathogens, the antibodies produced by the mother will be more beneficial to the baby than any other mammal on earth;
Human milk contains a variety of chemokines and cytokines that can cross the intestinal barrier, where they communicate with baby's cells to influence immune activity. Many cytokines and chemokines have multiple functions, and milk-borne cytokines may be grouped broadly into those that enhance inflammation or defend against infection, and those that reduce inflammation (Ballard &amp; Morrow, 2014);
Bifidus factor: promotes the growth of beneficial bacteria in the gut and limits the growth of disease-causing bacteria (source: NIH's MedLine).
A variety of immune cells, including macrophages, T cells, stem cells, and lymphocytes. In early lactation, the breastfed infant may consume as many as 1010 maternal leukocytes per day. They can differentiate into dendritic cells that stimulate infant T-cell activity (Ballard &amp; Morrow, 2014).
Breast milk changes in composition according to the infant's needs, something cow's milk can never accomplish; The first fluid produced by mothers after delivery is colostrum, which is distinct in volume, appearance and composition. Colostrum is produced in low quantities in the first few days postpartum and is rich in immunologic components such as secretory IgA, lactoferrin, leukocytes, as well as developmental factors such as epidermal growth factor. It contains relatively low concentrations of lactose, indicating its primary functions to be immunologic and trophic rather than nutritional. Levels of sodium, chloride and magnesium are higher and levels of potassium and calcium are lower in colostrum than later milk (Ballard &amp; Morrow, 2014). 


For more complete reviews on the immunochemistry of breast milk, please refer to Ballard &amp; Morrow (2014) and Mantis et al. (2011)

The mother also benefits:


Breastfeeding burns extra calories, so it can help mothers to lose pregnancy weight faster; 
Breast feeding releases the hormone oxytocin, which helps the uterus return to its pre-pregnancy size and may reduce uterine bleeding after birth;
Breastfeeding lowers the risk of breast and ovarian cancer. It may lower the risk of osteoporosis;
Breastfeeding has been linked to higher IQ scores in later childhood in some studies (source: WebMD);
During lactation, menstruation ceases, offering a form of contraception;
Mothers who breastfeed are less likely to develop breast cancer later in life;
Breastfeeding is more economical than formula feeding;
Hormones released during breast-feeding create feelings of warmth and calm in the mother (source: NIH's MedLine).


In terms of bonding:


Skin-to-skin contact during breastfeeding, and eye contact all help a baby bond with the mother and feel secure. 
Oxytocin release is known to promote the bonding with bubba. A newly born person can cause dramatic changes in the life of mommie (and daddy!), and a steady release of oxytocin promotes the bonding and love of mommie for the bubba (source: WebMD).


References
- Ballard &amp; Morrow, Pediatr Clin North Am (2013); 60(1): 49–74
- Mantis et al., Mucosal Immunology (2011) 4: 03–611
- Quintanilla, Nurs Times, (1996); 92(31): 35-7