Is having a dead nerve or infected pulp the same thing as an abscess tooth? thank you!

Basically yes : The nerve ina tooth can be dead but not acutely infected and paainful. Even if there is no pain and swelling, these teeth still need to have root canal.

Somewhat: The nerve and the blood vessels within the pulp chamber and canal(s) of the tooth can be non-vital (dead) and not be acutely infected. An abscess is an acute infection. Chronic infections can become acute. In all cases, both chronic and acute, these teeth require antibiotics and root canal therapy or an extraction.

On its way: Once the nerve dies either by decay or trauma bacteria invades the nerve. Radiographically sometimes you can see the infection, sometimes the patients do not even know they have it. When the bacteria becomes active & is proliferating it looks for an escape as the body is sending cells to combat the infection. This creates intense pressure once bacteria has eaten through bone an abscess develops.