Hasan Ibn Abbas Marufi wrote a letter to Imam Reza a.s. (the 8th Imam) and asked this question. so Imam wrote in response:

> عَلِيُّ بْنُ إِبْرَاهِيمَ عَنْ أَبِيهِ عَنْ إِسْمَاعِيلَ بْنِ
> مَرَّارٍ قَالَ كَتَبَ الْحَسَنُ بْنُ الْعَبَّاسِ الْمَعْرُوفِيُّ إِلَى
> الرِّضَا ع جُعِلْتُ فِدَاكَ أَخْبِرْنِي مَا الْفَرْقُ بَيْنَ
> الرَّسُولِ وَ النَّبِيِّ وَ الْإِمَامِ قَالَ فَكَتَبَ أَوْ قَالَ
> الْفَرْقُ بَيْنَ الرَّسُولِ وَ النَّبِيِّ وَ الْإِمَامِ أَنَّ
> الرَّسُولَ الَّذِي يُنْزَلُ عَلَيْهِ جَبْرَئِيلُ فَيَرَاهُ وَ يَسْمَعُ
> كَلَامَهُ وَ يُنْزَلُ عَلَيْهِ الْوَحْيُ وَ رُبَّمَا رَأَى فِي
> مَنَامِهِ نَحْوَ رُؤْيَا إِبْرَاهِيمَ ع وَ النَّبِيُّ رُبَّمَا سَمِعَ
> الْكَلَامَ وَ رُبَّمَا رَأَى الشَّخْصَ وَ لَمْ يَسْمَعْ وَ الْإِمَامُ
> هُوَ الَّذِي يَسْمَعُ الْكَلَامَ وَ لَا يَرَى الشَّخْصَ 
reference:
[الكافي][1] ج 1، ص176،باب الفرق بين الرسول و النبي و المحدث


According to this hadith and also other references Some differences of prophet and Imam are:

 1. Imam does not have The Tashree' authority (i.e. legislation or
    bringing a new religion making rules of religion) but prophet have.
    Imam only protects the religion of prophet from going astray but
    does not have the right to make any change or new law in religion.
    prophet transfers all the divine knowledge of religion to next Imam
    (prophet is both Imam and prophet at the same rime like Quran says
    about Ibrahim a.s.) and so Imam has the perfect knowledge of
    religion. for example Imam can not change halal and haram.
 2. prophet can see revelation angel Gabriel  directly. but Imam does not see
    angel directly but the messages of God is inspire to him or Imam
    only hears angle (but not seeing)
 3. prophet can see unseen universe directly but Imam learn about unseen
    from prophet.
 4. according to this verse rank of Imam is higher than prophet:

> وَإِذِ ابْتَلَى إِبْرَ‌اهِيمَ رَ‌بُّهُ بِكَلِمَاتٍ فَأَتَمَّهُنَّ
> قَالَ إِنِّي جَاعِلُكَ لِلنَّاسِ إِمَامًا قَالَ وَمِن ذُرِّ‌يَّتِي
> قَالَ لَا يَنَالُ عَهْدِي الظَّالِمِي
> 
> And remember that Abraham was tried by his Lord with certain commands,
> which he fulfilled: He said: "I will make thee an Imam to the
> Nations." He pleaded: "And also (Imams) from my offspring!" He
> answered: "But My Promise is not within the reach of evil-doers."

http://tanzil.net/#2:124

It can be understood that the rank of Imam is higher than prophet because prophet Ibrahim a.s. could become Imam after passing exams and it was at ends of his life because he asks "and from my children"? we know he did not have children until he was aged according to other verses. 

please note this does not mean the rank of 12 Imams of shia like for example Imam Ali a.s. was higher than prophet because prophet Muhammad PBUH was prophet and Imam at the same time and his rank is higher than all 12 Imams. 


  [1]: http://en.wikipedia.org/wiki/Kitab_al-Kafi