Obviously Jesus and "my god" are different beings. Jesus is the son of the Christian God. "My god" can be any pagan god: it could be a plant in your back garden that your grandmother worships. Jesus is not the son of the plant.
I think people say, "Oh Jesus" and "Oh my god" to express alarm, surprise, distress, joy, ecstasy, disgust and all kinds of emotions. In this regard, there is no material difference between the two.
I am a little worried about you. You should not be watching a lot of TV dramas where people say "Oh Jesus" and "Oh my god" a lot. It's very bad for your language, and perhaps even your social etiquette. Perhaps it is much better to watch members of the British royal family pay personal tributes to the Queen over family cine films from as early as when King George VI was still the Duke of York, or Brian Sewell on art and civilisation, or Brideshead Revisited from 1980, or vintage Julie Andrews shows, or... There are so many other things that you can watch where the English is so much better - in fact, the best in the world.