There is not much difference between the two. Modern Chinese uses a lot of two character words, mainly for balance. If you said wo xueguo zhongwen (I have studied OR I did study Chinese), then dropping xi would be OK as guo would appropriately balance with zhongwen.

xuéxí means to study and practise however, this applies to things learning the piano, martial arts because you need to practise in order to learn. However, in things like medicine, you can only learn because you cannot practise them without a licence. For example, learning the martial arts, you just say xué wu. To learn and practise the martial arts, then xuéxí wu yi is OK but not xué xí yī xu to learn and practise medicine.

I think JessieCheng explained well and I would like to attach more. The sole characters "xue" and "xi" have slight difference in meaning. "xue" is more likely to refer to the state that you are initially exposed to something and learn it, "xi" is more likely that you have been in the process of learning and learning by practicing and repeating. When you use "xue", there is often a noun after it specifying what you learn, and the noun is what you most want to stress in the whole sentence. When you use "xue xi", you are stressing your current state, that you are not playing or doing somethine else. In addition, "xue xi" sounds more formal than "xue“, "xue" is more often used orally than "xuexi".

In the ancient Chinese language,the xúe means learn and the xí means review after learning.Today,we say xuexi means to learn,and xue means learn sth..If you learn some knowledge and you dont want to forget,you could say 温习,not 学习.

Claude gave a veery good answer. The 'inner' grammar of modern Chinese is rather subject to the prosody. I had talked about this detailedly at italki.com before for several times. And I'd like to add that, when a monosyllable word and a bisyllable word(which has the same character as the fommer word) have the same meaning, they often differ in the style or register. The one-character word is usually more colloquial and the other more formal. There arre lots of instances of the case. And sometimes even both words are formal, the latter is a bit more formal: 使VS使得.