DifferentialEquations.jl can definitely be used to do circuit simulations but you can’t (today) import a netlist and instantiate models of ICs by part number and be off and running like you can with a SPICE program. Instead you need to be able to write your equations and do a bit of algebra up front and in back to get all the answers.
On the other hand, you can describe a modest amount of circuitry that way, along with equations for some physical things, and simulate control algorithms using callbacks (which can even call C code if you wish), letting you do things that are difficult or impossible with SPICE.