Lightspeed is a websocket server.
Livewire talks to the server over normal HTTP requests and for anything pushed from the server it uses Laravel Echo. Lightspeed is a Pusher-compatible server so you'd point Echo at it like you would Reverb.
The thing Lightspeed adds is letting the browser send messages back into Laravel over that same socket, with auth on each one. Livewire doesn't do that its actions are still HTTP.
For most applications Livewire would be the goto. Lightspeed is kind of a high speed specialist option for building real time apps like you might do with node.js for exmaple.