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These two techniqies seem closely related:
Are the mathematics the same, just different communities (math or stats), like in Tikhonov regularization or ridge regression?
Or is there a difference, e.g., that IRLS neglects correlations (after weighted least squares) while FGLS does not (after generalized least squares)?
No, IRLS is a strategy for solving more general p-norm minimization problems by means of a sequence of related 2-norm (least squares) problems.
FGLS may be specialized for a diagonal covariance matrix in Feasible Weighted Least Squares (FWLS).