That part of the paper is showing why the efficient frontier is the same regardless of whether you are maximizing utility, maximizing returns given variance, or minimizing variance given returns.
Inequality constraints tend to be a bit more work to deal with analytically, so that might be a reason why they use the equality constraint on one of them.
Usually you get the same portfolio regardless of whether you use the inequality constraint or the equality constraint. I don't think this paper is really trying to evaluate equality vs. inequality or anything. They're focused more on the general point. The inequality constraint is just telling you that the variance must be less than some amount. The equivalent when minimizing variance given returns would be an inequality constraint such that the return of the portfolio is greater than or equal to some amount.