Formula feeding increases risk of baby developing type I (juvenile, insulin-dependent) diabetes.
The results of a study in Finland suggest that young age at introduction of dairy products (formula is a dairy product) and high milk consumption during childhood increase the level of cow's milk antibodies in the children's systems. This factor is independently associated with increased risk of insulin dependent diabetes. Virtanen et al: "Diet, Cow's milk protein antibodies and the risk of IDDM in Finnish children." Childhood Diabetes in Finland Study Group. Diabetologia, Apr 1994, 37(4):381-7
“The longer a child is breastfed, the less likely he/she is to be diabetic. Children breastfed one year or longer have half the risk of being diabetic of those breastfed for less than one year.”
Mayer, EJ, Hamman RF, Gay EC, et al. "Reduced risk of IDDM among breast-fed children". Diabetes, 1988;37:1625-1632
Virtanen SM, Rasanen L, Aro A, et al. "Infant feeding in Finnish children <7 yr of age with newly diagnosed IDDM".
Diabetes Care, 1991;14:415-417
Gerstein HC. "Cow's milk exposure and type 1 diabetes mellitus". Diabetes Care. 1994;17:13-19
Borch-Johnson, K., et al., "Relation between breastfeeding and incidence of insulin-dependent diabetes mellitus". Lancet 2:1083-86 (1984)
Karjalainen J et al, New Eng J Med, 1992 327:302-7

Formula feeding increases baby's risk of otitis media (ear infections)
"Otitis media is up to 3-4 times more prevalent in formula-fed infants". Aniansson G, Alm B, Andersson B, et al. "A prospective cohort study on breast-feeding and otitis media in Swedish infants". Pediatr Infect Dis J. 1994; 13:183-188
Kovar MG, Serdula MK, Marks JS, et al. "Review of the epidemiologic evidence for an association between infant feeding and infant health." Pediatrics. 1984:74:S615-S638
Saarinen UM. "Prolonged Breast Feeding as prophylaxis for recurrent otitis media." Acta Paediatr Scand. 1982;71:567-571

Formula fed babies are more likely to get eczema
"We conclude that breastfeeding is prophylactic against atopic disease, the effect extending into early adulthood. Breastfeeding for longer than 1 month without other milk supplements offers significant prophylaxis against food allergy at 3 years of age, and also against respiratory allergy at 17 years of age. Six months of breastfeeding is required to prevent eczema during the first 3 years, and possibly also to prevent substantial atopy in adolescence." The differences by infant feeding method were so pronounced that they "suggested an influence of early milk feeding that may exceed the heredity burden.
Saarinen UM, Kajosaari M. "Breastfeeding as prophylaxis against atopic disease: prospective follow-up study until 17 years of age." Lancet. 1995; 346:1065-69.
Eczema was less common and milder in babies who were breast fed (22%) and whose Mothers were on a restricted diet (48%). In infants fed casein hydrolysate, soymilk, or cows milk, 21%, 63% and 70% respectively, developed atopic eczema. Chandra R.K., "Influence of Maternal Diet During Lactation and the Use of Formula Feed an Development of Atopic Eczema in the High Risk Infants". Br Med J. 1989

Formula feeding increases chances of baby developing allergies
Lucas A, Brooke OG, Morley R, et al. "Early diet of preterm infants and development of allergic or atopic disease: randomized prospective study". Br Med J. 1990:300:837-840
Halken S, Host A, Hansen LG, et al. "Effect of an allergy prevention programme on incidence of atopic symptoms in infancy". Ann Allergy. 1992;47:545-553
Saarinen UM, Kajossari M. "Breastfeeding as prophylaxis against atopic disease: prospective follow-up study until 17 years old."
Lancet. 1995;346:1065-1069

Risk of Sudden Infant Death
“In Scotland, 80% of cot death victims are artificially fed, while only 56% of normal infants are artificially fed.”
Bartholomew S and MacArthur BA. Comparison of infants dying from sudden infant death syndrome with matched live controls. Social Science & Medicine 1988 27(4):393-97
Ford et al, Intl J Epidemiol, 1993 22:885-9 see also Gilbert et al, BMJ, 1995 310:88-90
(Frederikson et al, Amer J Dis Child. 147:460 Abstract) Klonoff-Cohen et al, JAMA, 1995 273:795-8

Formula fed babies have poorer social development
The psychomotor and social development of breastfed babies clearly differs from that of bottlefed ones and leads at the age of 12 months to significant advantages of the psychomotor and social capabilities.
Baumgartner, C.,"Psychomotor and Social Development of Breast Fed and Bottle Fed babies During their First year of Life".
Acta Paediatrica Hungarica, 1984

Formula Feeding is associated with lower I.Q.
The latest study to support this statement was done in New Zealand. Here an 18 year longitudinal study of over 1,000 children found that those who were breastfed as infants had both better intelligence and greater academic achievement than children who were infant-formula fed. Horwood and Fergusson, "Breastfeeding and Later Cognitive and Academic Outcomes", Jan 1998 Pediatrics Vol. 101, No. 1
Morrow-Tlucak M, Haude RH, Ernhart CB. Breastfeeding and cognitive development in the first 2 years of life. Soc Sci Med. 1988:26;635-639
Lucas A., "Breast Milk and Subsequent Intelligence Quotient in Children Born Preterm". Lancet 1992;339:261-62
Wang YS, Wu SY. The effect of exclusive breastfeeding on development and incidence of infection in infants. J Hum Lactation. 1996; 12:27-30

Breast milk lowers risk of baby developing asthma
Breastfed babies have lower risk for developing recurrent wheezing when they are older (age 6 or more). Archives of Pediatric and Adolescent Med., July 1995

Formula fed babies have a higher risk of developing certain childhood lymphomas
Davis MK, Savitz DA, Graubard BI. "Infant feeding and childhood cancer." Lancet. 1988;2:365-368
Shu X-O, Clemens H, Zheng W, et al. "Infant breastfeeding and the risk of childhood lymphoma and leukaemia". Int J Epidemiol. 1995;24:27-32

Babies fed formula have more gastro-intestinal infections
“In the UK, babies breastfed for 13 weeks or more were 3 times less likely to contract gastrointestinal illness regardless of socio-economic conditions.”
Howie et al, BMJ 1990 300:11-16

This list is by no means complete but offers some of the health differences between breastfed and artificially fed children for your information. Further risks to both infant health and lifelong health are being found by research all the time.
Further reading:
"Breast Is Best" by Stanway & Stanway
http://www.promom.org/101/