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They mean the same thing. Square meters is the same as meters squared.
A linear (lineal) metre is the same thing as a metre. I think it's a dumbed down term for a metre so people won't get confused with metre, square metre and cubic metre.
None! They both mean the same thing. For example, if your floor measures 3 meters long and 2 meters wide you would need 6 square meters of carpet. Alternatively you could say "My floor measures 6 meters squared." Five meters squared. This refers to a linear measure, five meters. It also tells what function to apply to the measure-- make a square with it. If I make a square with a five meter chain, I will end up with 25 square meters. Five square meters. This will be equivalent to five of the 25 squares that the above area contains. Quite different. If you have a floor measuring 3 meters by 2 meters and you ask the carpet guy for carpeting to cover 6 meters squared, you will get an unpleasant surprise.
Nothing - they're both the same measurement. You can use either - depending on the circumstances. For example, you could say a room is 20 square metres in area, or it is 20 metres squared. Either way is correct.
None because they are both describing the same thing in square cm