Assuming you are not going from (say) adsl to fibre, (or visa versa) then you can use the same connection. However, if the existing cable has been there a few years, you may get a faster connection using the new one - copper does deteriorate with time. As for the power supply, you need to make sure it's the same voltage out and at least the minimum power output. Normally, different voltages are a different size connector, but make sure. Plugging in the wrong voltage can make for an expensive mistake.
Samsung Galaxy A9 review: Hands-on