Time Domain solutions with differential equations give you a complete solution for an electrical circuit starting with some initial conditions usually. Time Domain solutions with a numerical tool like SPICE or EMTP produce the same results using numerical integration - and are preferable for problems larger than the typical simple circuits of textbooks.
Phasor Domain solutions are useful if you care about a particular steady-state of a circuit driven by AC sources. A good example of this are short-circuit programs used by utility protection engineers like CAPE, ASPEN, or DigSILENT. These are phasor based programs and typically are setup to provide subtransient (first cycle or so after a fault occurs), transient (up to around 20-30 cycles), and synchronous short-circuit voltages & current phasors. The bulk of protective relays used in the utility system are phasor based or RMS based elements. Here is an SE question I answered and shows use of phasors for a directional element.
Laplace Domain, as @jonk mentions in his comment, is an easy way to solve differential circuit problems (but not too large as it gets unwieldy) with the result being closed-form solutions for quantities of interest. Here are a few SE posts I have answered that can help you in this regard: LC circuit, LC circuit with driving functions, and a neat RC circuit.