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I read in some good grammar book that
"Above" and "over" can both mean "higher than":
The example from the book
Can you see the helicopter above/over the palace?
Then below it was written that
We prefer "above" when one thing is not directly over another
The example from the book
We've got a little house above the lake
The question is is the helicopter in the first example directly over the palace or no if both "above" and "over" can be used?
In the helicopter sentence, using one word or the other doesn't change the meaning of the sentence.
The book's statement that we prefer "above" when one thing is not directly over another is not a real rule, and not terribly important to remember. It's true that the two words are used just slightly differently—"over" for something directly over another and "above" for something merely at a higher level than another—but it's not a big deal and in general you can use them interchangeably.
"We've got a little house over the lake"
doesn't sound too different to me than
"We've got a little house above the lake"
I think they actually both sound a little awkward. I'd say "We've got a little house by the lake" or "overlooking the lake".
Depending on the context and tone, "above" and "over" could very well change the meaning of the sentence. Above really only has the one meaning, while over can, with some nouns, mean "on the other side of" as well as being a synonym of above. To modify the helicopter example, "There is a helicopter above the hill" can really only mean there is a helicopter in the sky above a hill. "There is a helicopter over the hill" could mean either that there is a helicopter in the sky over the hill, or that there is a helicopter on the far side of the hill, perhaps even on the ground. Fortunately, a native speaker just wouldn't ask "can you see the helicopter over the hill?" if one meant "on the other side of the hill." However, the question might be about whether or not the hill impedes 'your' view of the helicopter, but in that case all parties would probably have already agreed upon which specific hill and helicopter everyone is talking about. And in either case, the majority of answer to either question would answer both ("Yes, I see it," "No, the hill is in the way," or "No, I'm blind" being just a few examples).
As for the lake house, neither "over" nor "above" can really be applied unless the house can hover, fly, or is in some other way suspended above/over the lake, in which case the speaker would know their house is unusual enough to be more specific. Tangentially, most houses are largely "above" ground, but never "over" the ground; a floating house (that somehow isn't also a boat) might be accurately described as "above the lake," but it would probably never be casually phrased that way. Colloquially, I would assume in the case of "over" that the speaker meant "on the other side of" (but would find it unusually enough to ask for clarification), and I would be incredulous of any claim of a house "over" the lake. A house "on" the lake is a common phrase to refer to a house very, very close to the lake, without any other property lines between it and the water. If there is a road between the lake and the house, it's not "on" the lake.
