The prepositions 'Across' and 'Over' are used with similar meaning, but there are differences.
Both Across and Over can be used to mean 'on or to the other side' of a line, river, road, and so on (position or movement related to things that are 'long and thin'.
Examples:
They walked across or over the road...
See if you can jump over or across the corridor...
Her room's just over or across the corridor...
They will be over/across the frontier by midnight...
Over is used for movements on or above water, but not in water. For example: How long would it take to swim across the river? (Not: ...over the river?)
Both Across and Over can mean 'on the other side of' a high barrier (like a hedge, a fence, a wall, a mountain range...), but only Over is used for a movement to the other side of something high. Compare the following:
If we can be over/across the fence before sunrise, we have got a chance.
When I last saw them, they were climbing very slowly over the fence. (Not: ...across the fence.)
Both Across and Over can be used for movement inside an area (Ex: fields, a desert, a dance floor). Note the following example: Who are those people wandering over/across the fields?
However, we only use across when we mean 'from one side to the other of the area'. Note the following example: It took them five weeks to walk across the desert. (Not: ...over the desert.)
However, Over is not normally used for movement in a 3-D space (like a room). Note the following example: He walked across the room, smiling strangely. (Not: ...over the room.)
Note that the adverb 'over' has a wider meaning than the preposition 'over'. You can't say: Let's swim over the river to the church, but you can say: Let's swim over to the church...
