Regular standard errors are biased when the data comes from a survey sampling design.
The article "Wait Wait, Don't Tell Me... You're Using the Wrong Proc!" explains that the bias is due to the violation of the independence assumption because the target population from which survey sample data is drawn is finite.
Example 4 of "Guidance for use of weights: an analysis of different types of weights and their implications when using SAS PROCs" simulates a sample of data that is 1% the size of the target population. The difference between the regular standard errors (WOLS/ML) and the survey standard errors (EE) is quite large.
Table 3 Monte Carlo mean, empirical, WOLS/ML and EE SE WOLS/ML EE Mean 9.14 9.14 SE (WOLS/ML): 1.36 SE (empirical): 0.726 SE (EE): 0.726 EE, estimating equation; ML, maximum likelihood; WOLS, weighted ordinary least squares.
With a target population so much larger than the sample, it seems like the infinite population assumption would be OK or that the two methods for computing standard errors would not be very different.
Do survey sample standard errors take into account/correct for anything else besides finite population?