I would have thought the spaceshooter pew, pew thing, was a great comparison to ED. Both are very pew, pew.
No, they're absolutely fundamentally different. In good ways.
In Elite, it takes you about half an hour of docking, taking missions, travelling and whatnot, before you're in any kind of action. Elite is essentially an MMO and your ingame character "lives" to some extend in this world. You spend dozens to hundreds of hours to upgrade and equip your ship and the world is mostly mapped out and connected.
Half an hour in Everspace is ridiculously more packed. You only move through levels, you fight bands of enemies, mine ores and gas, upgrade your ship systems, listen to your pilot's banter with the ship AI, defend or attack freighters, die two or three times, upgrade the ship with the cash you earned playing after each death, and get teleported into the action again right away. The world is connected through jump points, is generated differently after each death based on level templates and you always try to reach the end, being on the run from a fleet that catches up and invariably (?) kills you if you linger too long. Ship physics are also far more strongly abstracted than what you have in Elite. You point somewhere with the mouse, your ship goes there and your weapons shoot there. It's twitch reaction based and you currently have only one ship available (that does everything).
I managed to give the current beta a short go yesterday and was rather pleasantly suprised. While I don't understand most of the systems, yet, the loop of slowly upgrading and being able to survive longer and longer and reach further sectors is oddly compelling.