They seem to be defined identically in this usage case (though 摔 has other meanings). Is there any difference in usage?
When 摔, 扔, 投 mean 'to throw':
摔 is a more forceful throw. For example, '把杯子摔在地上' (you throw the cup onto the ground with considerable force)
扔 is a more generally directed throw. For example, '把垃圾扔在地上' (you just throw the garbage onto the ground, doesn't matter which direction)
投 is a more specifically aimed throw. For example, '把篮球投入篮中' (you aim and throw the basketball into the basket)