I don't understand the difference between these two prepositional verbs. In examples:
Mein Freund kann mir am Sonntag beim Umziehen helfen.
Kannst du mir mit den Hausaufgaben helfen?
What you are helped with, follows bei and mit. The only difference I see, is that bei used with 'verbal nouns' or gerunds in English.
You often have situation where more than just one preposition fits. You find such cases in English as well as in German.
All this examples are correct (I split »beim« into »bei« and »dem« to have the preposition bei separated from the article dem. Read more in the addendum.):
Mein Freund kann mir am Sonntag bei dem Umziehen helfen.
Mein Freund kann mir am Sonntag mit dem Umziehen helfen.Kannst du mir bei den Hausaufgaben helfen?
Kannst du mir mit den Hausaufgaben helfen?
Both versions have very similar meanings, but not exactly the same.
If you use »helfen bei etwas«, then this etwas/something is a thing or a task where you want that the helpers does some work at this thing. So the thing is here seen as a passive object that affords to be modified by the helper.
If you say »helfen mit etwas«, then this etwas/something is a thing or a task where you already are involved (or will be involved soon), and you (not the thing) need support to be able to handle this thing or task.
So »bei« sets the focus on »Umziehen« or »Hausaufgaben«. They are the things or tasks that the helper should support. »Mit« sets the focus on the person that needs help with this things or tasks.
addendum about »bei dem« vs. »beim«:
»Beim« is a short form of »bei dem« like »don't« is a short form of »do not«. So, theoretically it is possible to split »beim« into »bei« and »dem« like I did here for didactical reasons. But in real life this is bad style. Better use »beim« when ever »bei« meets »dem«.
Addendum 2:
There is no standard German short form for »mit + dem«. But in some dialects you can hear »mitm« instead of »mit dem«. Sometimes, when speakers have a sloppy pronunciation, they even omit the t and you hear only »mimm«. You should know what »mitm« (or »mimm«) means when you hear it, but you better don't use it, since it is wrong in standard German.
