Let $B$ be a three-dimensional Brownian motion. We compare two regularizations of its self-intersection local time: a temporal cutoff excluding pairs of times separated by less than $\varepsilon$, and a spatial regularization obtained by replacing the Dirac mass with the heat kernel $p_\varepsilon$. We prove that the difference between these regularizations converges to a non-degenerate centered Gaussian random variable as $\varepsilon\to 0$, with convergence of the corresponding variances. The limiting variance is identified by an explicit $L^2$-calculation. We also briefly discuss how the comparison of regularization schemes arises naturally in the analysis of the three-dimensional Edwards model.