I found a solution to this online already but I'm curious to see why my method specifically doesn't work.
Let $X_1,X_2$ be independent r.v.s with parameter $\lambda$.
Then let $T = X_1 - X_2$.
We have
$$\mathbb{P}(T\leq t) = \mathbb{P}(X_1 \leq t + X_1) \\ =
\int_{0}^\infty \mathbb{P}(X_1\leq t + X_1 | X_1 = x) f_{X_1}(x)\ dx\\
= \int_{0}^\infty \mathbb{P}(X_1 \leq t + x | X_1 = x)f_{X_1}(x) \ dx \\
= \int_0^\infty \mathbb{P}(X_1 \leq t + x)f_{X_1}(x) \ dx \\
= \int_0^\infty (1- e^{-\lambda (t+x)})f_{X_1}(x) \ dx \\
= \int_0^\infty f_{X_1}(x) - \lambda e^{-\lambda x} e^{-\lambda t - \lambda x}\ dx \\
=1 - \int_0^\infty \lambda e^{-2\lambda x}e^{-\lambda t}\ dx \\
= 1 - e^{-\lambda t}\frac{1}{2}\int_0^\infty 2\lambda e^{-2\lambda x} \ dx \\
= 1 - \frac{1}{2}e^{-\lambda t}$$
Differentiating, I get that
$$f_T(t) = \frac{\lambda}{2}e^{-\lambda t}
$$
which doesn't seem like a density..