Provided that the specification is right, will these two different ways of removing the cross sectional bias result in the exactly same result (contrasting to first difference which will result in different coefficients)? Here is how I have defined the terms, just to be clear:
Fixed effects with demeaning = demean variables and run regression on the demeaned data to calculate the coefficient.
LSDV = Create dummy variables for each cross section except for one and run the regression to calculate the coefficient.