So there is no death penalty?
There is an insurance fee that has to be paid plus the cost of modules when you die. If you bring up the systems screen on the right (4 key by default) it will show your ship insurance cost which is just for the base ship (no modules) if my memory is correct. It can be skipped and you can go back to the starter ship or declare bankruptcy too as a last resort (starts the save over I think).
That is if you were asking about the death penalty in general. If you meant between Open/Solo there isn't anything different that I know of and the game just has that one death/insurance system outlined above.