Exclusivity is not a good thing, at least as far as youth sports are concerned.
While there are arguments to be made for concentrating on one sport only, the case is more persuasively made for encouraging young athletes to participate in multiple sports. There may come a point in time that it is the right thing to do for a particular athlete, but it should be the exception rather than the rule and should happen later rather than sooner. This opinion, although I do not expect all will agree with it, is based on my experience as both a coach and a parent.
News flash for my fellow parents: Very few of our children will be collegiate athletes, let alone professionals. Don't get me wrong, there are those who are truly gifted with the physical, mental and emotional attributes that will allow them to succeed at the highest level but, as I will explain later, even those outliers are at risk. While occasionally it is the child that wishes to play one sport exclusively, it is my perception that typically a parent, sometimes at the urging of a coach, is driving force behind the push toward specialization.
The Advantage [Please note that the title of this section is intentionally written in the singular] :
Specific technical skill development. That's it. I don't believe that I could argue that the constant repetition of a skill critical to the success at a particular sport does not allow an athlete to perform better. Most, if not all, coaches have espoused the maxim that the more "touches" a young player can get in their chosen sport, the greater the likelihood that the athlete will become particularly adept at that skill. But at what cost?
The Risks [Here the plural is intentional] :
• Physical and Mental Health
Studies have shown that for most sports, there is no evidence that intense training and specialization before puberty are necessary to achieve elite status. Risks of early sports specialization include higher rates of injury, increased psychological stress, and quitting sports at a young age.  Some of the negative consequences of specializing in one sport too early are overuse injuries and chronic injuries such as tennis elbow, rotator cuff injuries, stress fractures, and ACL injuries, especially in female athletes. Early specialization also contributes to a one-dimensional self-concept as a result of “a constrained set of life-experiences”. 
If you need more concrete proof, consider the following :
1. Children who specialize in a single sport account for 50% of overuse injuries in young athletes according to pediatric orthopedic specialists.
2. A study by Ohio State University found that children who specialized early in a single sport led to higher rates of adult physical inactivity. Those who commit to one sport at a young age are often the first to quit, and suffer a lifetime of consequences.
3. In a study of 1200 youth athletes, Dr Neeru Jayanthi of Loyola University found that early specialization in a single sport is one of the strongest predictors of injury. Athletes in the study who specialized were 70% to 93% more likely to be injured than children who played multiple sports!
4. Children who specialize early are at a far greater risk for burnout due to stress, decreased motivation and lack of enjoyment.
5. Early sport specialization in female adolescents is associated with increased risk of anterior knee pain disorders including PFP, Osgood Schlatter and Sinding Larsen-Johansson compared to multi-sport athletes, and may lead to higher rates of future ACL tears.
• Lack of Exposure...
...to other sports. By specializing too early the young athlete is being deprived of the opportunity to discover other sports that may be more enjoyable to them or that may actually be their forte. What happens when the rest of the children catch up in size to your son who figured was a shoe-in for a free ride to Notre Dame to play linebacker? Or your fleet-footed daughter who was her team's star striker until the rest of the girls developed speed.
...to other children. By playing sports with the same small group of children year-round limits the chances to develop meaningful friendships outside of this "inner circle". Social life tends to revolve around the sport and its limited participants.
... to other coaches. One of the best things about coaching is that you can make an impact on someone's life years down the road. It's not by teaching "x"s and "o"s, but by teaching how to lead and to be a team-player, skills that will benefit a child in all areas of their life. Let's face it, some coaches do that better than others. By specializing too early you are limiting the collection of coaches that may have a positive influence on your child and teach them the skills that will help them succeed not only on the playing field, but in the classroom and the workplace.
The Experts Agree
Ryan Boyle, a six-time MLL all-star and three-time Team USA attackman, the co-founder and CEO of Trilogy Lacrosse and an ESPN College Lacrosse analyst, recently wrote: "It is true that multi-sport athletes have less time for lacrosse-specific training. But the short-term advantages of lacrosse-specific training pale in comparison to the long-term advantages of multi-sport participation. College coaches prioritize potential over polish. They believe multi-sport athletes have barely tapped into their lacrosse potential, whereas the specialists have already maxed out."
The college coaches will tell you that they concur. In a recent article in Lacrosse Magazine Chris Bates (Princeton), Matt Kerwick (Cornell), Janine Tucker (Johns Hopkins), Jeff Tambroni (Penn State) and Scott Marr (Albany) agree that multi-sport prospects tend to be more well-rounded athletes who have an ability to accept different coaching styles and understand the dynamics of being on a team. The coaches believe that by participating in different sports athletes develop a higher overall "Athletic IQ" and that being a member of different teams provides opportunities to develop game instincts that produce more athletic players. These coaches practice what they preach. Each has at least one child participating in multiple sports.
A few other coaches with just a modicum of success have also chimed in on the issue:
"There is nothing you can be doing in lacrosse on your own in the fall that would be better for you than going to football or soccer practice every day. You can go bang a ball against a wall all you want, but how do you become a better team player? By playing other team sports." ~ Dom Starsia (Virginia)
"Becoming coachable, paying attention to detail, understanding the importance of preparation, working toward a goal, understanding your role and evaluating your performance from playing other sports are such a huge advantage." ~ John Danowski (Duke)
As parents, let's let our kids be kids. Keep them active, but give them the opportunities to experience what the world of "play" has to offer. Guide them, but let them choose what make them happy, even if it's just running around in the playground. You had your turn, let them have theirs.
As youth coaches we have to encourage our players to participate in the sports that they enjoy and to give it their all. While you may not have your star lacrosse player during soccer season, you will have a better athlete when lacrosse season rolls around. Let's not force our best players to commit to only one sport, it is extremely short-sighted and ultimately not in the best interest of those we have given so much time and energy to help succeed, not just on the playing field but in all their endeavors.
In both roles, we are doing this for the kids, not for ourselves.
 Edward M. Wojtys, MD, Sports Specialization vs. Diversification, Sports Health. May 2013; 5(3): 212–213
 Balyi Way and Higgs, Long-Term Athlete Development, 2013
 http://www.laxmagazine.com/high_school/022714_boyle_quitting_sports_to_focus_on_one_is_ short -sighted