In the 18th century children's clothing resembled their parents for the most part, just like today's child wears jeans a t-shirt just like the adults.
But there were also styles that were exclusive to children that adults did not wear, just as today's adult does not wear onesies. (Okay I know that a woman's body suit is very similar to a baby's onesie, and yes you can find rompers made for adult too, but as a basic rule, they are not the staples of an adult wardrobe as much as they are of the infants wardrobe.)
Little boys would wear dresses until they were breeched which occurred between the ages of 2 and 7 years. boys dress style is slightly different from girls dress style. the main difference could be seen in hair and headwear.
Why did little boys wear dresses? Well the bigest practicle reason is for the ease of changing a diaper, and later potty training. In the later half of the century a new style for boys came about. these were called Skeleton Suits and the pants buttoned on to the top. I would guess that these were for boys who were fully potty trained as there are a lot of buttons to undo to get the pants off.
Girls dresses differed from women's in the fact that a girls dress closed in the back, where as most women's gowns closed in the front. Leading strings or ribbons which hang from the shoulder are also the mark of a young girl's dress.
For more information see Colonial Williamsburg's article on children's clothes
Childrens Fashions 1700-1750 on Wikipedia
Childrens Fashions 1750-1795 on Wikipedia