These two words are often confused. Alternate(ly) means by turns. Alternative(ly) means in a way that offers a choice. "The journey may be made by rail or alternately by road" means, if it means anything, that every other journey may be made by road. It does not mean, as the writer intended, that for every journey the traveler has a choice between the two means of transport. Conversely "alternatively they sat and walked in the moonlight, talking of this and that" cannot have been intended to mean that they sat and walked in the moonlight as an alternative to doing something else; what must have been intended is that they sat and walked alternately. Alternate can also be a verb meaning, in popular language, to "take it in turns".
See also Alternative.