Es lo mismo en muchos casos, puedes ver sus definiciones según la Real Academia de la Lengua Española http://lema.rae.es/drae/?val=aceptar http://lema.rae.es/drae/?val=admitir

They mean absolutely the same.

They're used in different circumstances but they mean the same thing. They're different words, that's it basically. Is like saying "I got admitted into a Special College" in that case, it can also be said "I got accepted into a Special College". Cuando admites algo, significa que estás aceptando algo que hiciste. Cuando aceptas algo, significa que diste tu aprobación para hacer algo. Basically, when you "admit" something, it means you're accepting something you've done. When you accept something, it means that you give your approbation to do something. (Nota: Es "Admitir" no "Adimitir").

En ingles: Are there any differences**

Es lo mismo :)