I am learning Arabic, and I was wondering if there is a difference.
I think they don't. When you say "Allahu Al-Akbar" this means that there is another Allah and this is unacceptable, because "Allah" is only one. But when you say "Allah Akbar" this means that Allah is greater than anyone or anything else and this is acceptable.
Allah Akbar means "Allah is the greatest". That's correct.
Allahu Al-Akbar means "Allah, who is great". That's wrong.
Maybe this would be a good Question for the Arabic SE proposal!
I'm no expert but to give you a short answer yes both would have at first sight the same meaning, but Allahu al-Akbar or Allah ul-Akbar الله الاكبر includes some kind of shirk because it's a comparisons as if you said Allah is greater then... so the listener would expect a deity/something/somebody with whom Allah would be compared in his greatness. But as is this "statement" wouldn't really make sense because something is missing! So it would be wrong as!
While Allahu Akbar الله اكبر is clearly saying: Allah is greater, and this is the most safe (from shirk) and only used expression in Islam.
And Allah knows best!
Yes, they are the same. It's just a matter of easing the wording for the English reader. The more correct wording is the second one Allahu Akbar, since it demonstrates that the Arabic word for Allah has a dhama at the end while Akbar has a sukun. The first wording "Allahu ul-Akbar" is almost incorrect since it implies that "ul" or the dhama is connected to Akbar rather than Allah.
I hope that makes sense, and Allah (SWT) knows best.