The equation describing the two-particle wave function for two identical fermions is given by
$\Psi^F(x_1, x_2) = \frac{1}{\sqrt{2}}(\psi_a(x_1)\cdot\psi_b(x_2) - \psi_a(x_2)\cdot\psi_b(x_1))$.
The term $\psi_a(x_2)\cdot\psi_b(x_1)$ has a negative sign, and the term $\psi_a(x_1)\cdot\psi_b(x_2)$ has a positive sign. How important is this sign convention?
The sign of the exchange term is arbitrary; it could be positive or negative. But the first term must always be positive.
It does not matter which term gets which sign; all that matters is that they have opposite signs. If you multiply a wave function by an overall factor of -1, you still obtain a valid wave function.
Both terms can have either sign, and each of the four sign combinations (++ , -- , -+ , +-) leads to a valid wave function.
These are the only signs possible for the two terms, because the second term is the exchange term and thus must have a negative sign.