Para aclarar un poco más, "adentro" se utiliza con verbos de movimiento, y "dentro" con verbos estáticos. Voy a -dentro. Estoy dentro.

I'm hardly trying to thing in a sentences that can show some difference between those words but nothing comes to mi mind... I think they are the same and they both can be used the same grammatically. "dentro" sounds a little more 'formal' or poetic for me though, one usually finds "dentro" on history/math/physics books or in certain kinds of literature. Meanwhile "adentro" is quite more coloquial and common between friends and family. If you have some examples of sentences with these words let me know and I'll try to tello you if they make sence.

It's the same