I know that the obvious differences are that prophets are made of clay and angels are made of Nur (light) and that angels have no free will while prophets do and prophets can die by the hand of humans while angels cannot. but what other differences are there?
Difference in the creation of Prophets & Angels
...and created him from clay.
In many verses the Quran is pointing at the fact that prophets and messengers are human, which seems to have been a problem or something amazing or astonishing for pagans and disbelievers. Therefore the Quran puts a emphasis on this fact: See for example (3:79, 10:2, 11:27, 16:103)
Quran on the other hand rejected and gave some logical explanation why prophets and messengers were human instead of being angles: saying even if that was the case disbelievers won't change their attitude 6:111 and saying that all former messengers and prophets were human and acted like any human being 25:20-23 and only those who don't believe in Allah and don't expect an afterlife would ask for messengers and prophets from Angels (see also 25:7). But if they were sent down to them that would mean a punishment from Allah.
Read also this hadith from sahih Muslim:
The Angels were born out of light and the Jinns were born out of the spark of fire and Adam was born as he has been defined (in the Qur'an) for you (i. e. he is fashioned out of clay).
So we learn that the the Angels have been made from light.
So angels are considered as infallible (always) worshiping creatures: no angel would want to disobey God, for not only would it make no sense, but it would contradict their nature.
The infallibility of prophets and messengers is based on verses like 2:136 and 2:285. Note that here sunni and shi'a differ, sunni say that this infallibility is restricted on the transmission of the message while shi'a consider their infallibility general.
Who are better?
Scholars have different opinions on this matter some say the angels are better than humans in general and prophets in special, others said prophets are better and superior to angels. Imam ibn Hajar al-'Asqalani quoted the evidences of both parties in fath al-Barry. Imam al Qurtoby said quoting some scholars:
قال بعض العلماء: لا طريق إلى القطع بأن الأنبياء أفضل من الملائكة أو أن الملائكة أفضل من الأنبياء، لأن طريق ذلك خبر الله تعالى أو خبر رسوله صلى الله عليه وسلم أو الإجماع، وليس ههنا شيء من ذلك،
which means there's no strong evidence neither in the Quran, nor sunnah nor Ijma' that may favor one of these opinions. Al-Qurtoby is a well known scholar of tafsir who was aware of many ahadith and apparently didn't seem to qualify the hadith of any of the opposite opinions as strong or stronger!
However you may find a kind of tendency in favoring the humans in this Arabic fatwa and the reference of my next quote:
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: Comparing angels to the righteous among mankind is a matter concerning which there are differences among the scholars, each of whom quoted texts to support his opinion. But the most correct view is that the righteous among mankind are superior to the angels on the basis of the final outcome because Allah, may He be glorified and exalted, will give them reward the like of which cannot be attained by the angels as far as we know. Rather the angels will be in the abode of the believers – namely Paradise – and will enter upon them from every gate, congratulating them (with the words): “Salamun Alaikum (peace be upon you) for that you persevered in patience! Excellent indeed is the final home!” [13:24 – interpretation of the meaning]. From the point of view of their origin, the angels are superior because they were created from light and their innate nature is to worship and obey Allah, may He be glorified and exalted, and they have been given the strength to do that. Allah, may He be exalted, says that among the angels are the angels of Hell:
“Over which are (appointed) angels stern (and) severe, who disobey not, (from executing) the Commands they receive from Allah, but do that which they are commanded” [66:6]
“And those who are near Him (i.e. the angels) are not too proud to worship Him, nor are they weary (of His worship).
They (i.e. the angels) glorify His Praises night and day, (and) they never slacken (to do so)” [7:206].
This is the decisive view concerning this matter. Indulging in discussion of this matter and seeking to determine whether the righteous among mankind or the angels are superior is a kind of superfluous knowledge. Man has no need of understanding and learning about this matter. End quote from Fataawa Noor ‘ala ad-Darb, (8/6). (Fatwa)
So as it seems that for a good comparison it needs to compare several Issues separately, as angels have been created from something considered as pure while human have been created from something which seems to be dirty or "worthless". On the other hand angels are not exposed to situation where they need to show their faith etc. while believers are ... while believers and messengers and prophets are always in a situation of being tested Angels have got their levels of heaven without a test. Therefore it is hard to compare from that point of view. But it seems fair to say that humans in general and prophets in special gained their level in Jannah due to their good deeds after a hard test.
See also Wikipedia about angels in Islam.