I would actually say that road bikes are going to be lighter than bmx most of the time. I have never compared the two so I may be wrong, but bmx bikes are actually quite heavy when you consider their size. BMX bikes are much smaller but they are built to take a beating and so they have very high spoke count wheels and very sturdy parts, which are heavy.
As for other differences, bmx bikes are desiged for track, park or free ride applications while a road bike is designed to cover a lot of distance on paved surfaces. You would not want to ride a bmx bike 100 miles (atleast I wouldn't), and a road bike wouldn't last very long if you started racing it on dirt tracks or messing around in street parks. Hope this helps.
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