It appears they are now made of the exact same substance gypsum (calcium sulfate) sidewalk chalk is sometimes dyed that is the only difference.

"Blackboard chalk is a substance used for drawing on rough surfaces, as it readily crumbles leaving particles that stick loosely to these surfaces. Although traditionally composed of natural chalk, modern blackboard chalk is generally made from the mineral gypsum (calcium sulfate), often supplied in sticks of compressed powder about 10 cm long"

"Sidewalk chalk is made of sticks of colored (and sometimes white) chalk (now mostly gypsum) used to draw on sidewalks, streets, and driveways. It is often done by children, but in many cities, talented adult artists create masterpieces on the walkways."
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There is really know difference in chemical make up, as both are made from calcium sulfate. The main difference is in size of the chalk, as sidewalk chalk is usually bigger so that it can be seen on the sidewalks, and that sidewalk chalk usually comes in colors. One is used for sidewalks, and the other is used for blackboards.

Sidewalk chalk are typically large colored (and sometimes white) sticks of calcium sulfate mostly used for drawing on pavement or cement sidewalk. It is sometimes used to draw a four square court or a hopscotch board.
Artists such as Kurt Wenner and Julian Beever have created intricate and realistic street paintings using the chalk and pastels. Sidewalk chalk is used at some universities to advertise for events, especially where there is much concrete. Prohibitions are set for where students can chalk, usually limiting it to areas that will be washed away with rain, or areas which are set to be cleaned of chalk markings.

Blackboard chalk (ie. regular white school chalk) is a substance used for drawing on rough surfaces, as it readily crumbles leaving particles that stick loosely to these surfaces. Although traditionally composed of natural chalk, modern blackboard chalk is generally made from the mineral gypsum (calcium sulfate), often supplied in sticks of compressed powder about 10 cm long.
Source(s):
http://en.wikipedia.org/wiki/Sidewalk_chalk
http://en.wikipedia.org/wiki/Chalk
http://en.wikipedia.org/wiki/Chalkboard

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As someone who has used a lot of both:

Chalkboard chalk seems to be both harder, but more brittle: it breaks more easily but doesn't crumble the way sidewalk chalk does. Both can be found in a variety of colors.
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