The widespread use of the unilateral Laplace transform reflects the fact that in practice we often deal with causal systems and signals that have a defined starting time (usually chosen as $t_0=0$).
The Fourier transform is mainly used for analyzing ideal signals and systems, such as ideal filters (e.g., low pass, high pass, etc.) and ideal signals such as perfect sinusoids. In these cases we have to deal with non-causal systems with impulse responses that extend from $-\infty$ to $\infty$. The same is of course true for sinusoidal signals or complex exponentials. Note that neither ideal filters nor the signals mentioned above can be treated by the Laplace transform.
One of the most important features of the unilateral Laplace transform is that it can be used to elegantly solve differential equations with initial conditions. The initial conditions are taken into account by the well-known differentiation property of the unilateral Laplace transform:
where $f(t)$ is a differentiable function, and $F(s)$ is its (unilateral) Laplace transform. The Fourier transform doesn't have an equivalent to $(1)$ for taking initial conditions into account. If the Fourier transform is to be used for solving a differential equation with non-zero initial conditions, then the initial conditions need to be modeled as additional sources.