So i'm no guru, but here's what i understand about the verb forms aforementioned.
I was talking to a japanese friend of mine and when I asked her this she asked me what the difference between "can" and "am able to" was. I think it is a similar situation in Japanese.
taberareru = I can eat
taberu koto ga dekiru = I am able to eat (The thing of eating I am able to do)
In terms of politeness, as long as you made taberareru into taberaremasu, it would be just as polite as taberu koto ga dekiru. (I guess in writing formally, you might put taberu koto ga dekiru). Now there are a few reasons your prof. might want you to say taberu koto ga dekiru (1. he might be old school and very formal (as most are) 2. He might not want you to confuse potentiality and passivitiy). I took some time to write up a little (or a lot) on the subject below, if you're interested take a read, it might help clear some more complicated questions up, but what is above is pretty much the simple answer to your question: "can" and "am able to".
So for a more detailed version:
taberu koto ga dekiru = taberareru (much more commonly heard is tabereru (take out the ra) which means the same thing. So for your japanese friends, most would say tabereru) - to make it more polite, seeing as it does end in ru, you can always add the masu stem. So taberaremasu (again, a mouthful, so taberemasu).
*Be careful though, ichidan verbs (those taberu=tabemasu) are not plug and chug. For example, miru (to see) becomes mieru and not mirareru (concerning potentiality! mieru = miru koto ga dekiru) There are a few of them that do this, it just takes some memorization and practice though.
For yodan verbs (those NOT taberu = tabemasu i.e. yomu, iku...etc) you can change the ending to e (yome, ike) and add ru on the end to express the same meaning.
yomeru/yomemasu = yomu koto ga dekiru
ikeru/ikemasu = iku koto ga dekiru and so forth.
Here is a page that might help a little http://www.timwerx.net/language/jpverbs/lesson51.htm
Laslty, the reason your professor might insist is because taberareru is technically the passive form of the verb. like "to be eaten." (see below if interested) It also is technically a more honorific form of the verb tabemasu (store clerks at times will use it, not as a potentiality, solely on honorific terms...weird right?) To be safe (and to keep your grade!) I would use the koto ga dekiru as your teachers requests (you may ask him/her if there is a reason behind this), but for normal life, you can just use tabereru.
Not to make it more difficult, but if you're interested.
taberu = (passive) taberareru 'to be eaten"
miru = (passive only!!) mirareru "to be seen"
kau = (passive) kawareru "to be bought"
yomu = (Passive) yomareru "to be read"
Sorry for the long response, I hope this helped![/i]