The differences between " मुझे (mujhē)" and " मैं (maiṁ)" in Hindi
In November 2010, I joined a course of Hindi language through some continuous education centre. Although the tuition fee is very inexpensive (18 hours at a rate of USD70), I unfortunately met one of the worst teachers ever through my life in learning foreign languages: Every time when I posted a question related in Hindi language (such as grammar, sentence structure, vocabulary, or even simply pronunciation), she either did not want to answer, or she answered with very little patience. Hence, I have a question in this language.

Here comes two examples:
1) I do not know.
2) I do not speak Hindi language.
As we see, both sentences have a common nominative subject, i.e., I.

Nevertheless, from the class and through the Google translator, I learnt that the above two examples in Hindi become
1)　मुझे नहीं पता (Mujhē nahīṁ patā)
2)　मैं हिंदी भाषा बोलते नहीं है (Maiṁ hindī bhāṣā bōlatē nahīṁ hai)

I hereby would like to know what might be the difference(s) between " मुझे (mujhē)" and " मैं (maiṁ)". Why do we have different representation of subject pronoun? In what situation(s) do we use which pronoun, respectively?

Thank you very much for your kind attention and any answers or comments, and good day to all.

Question

The differences between " मुझे (mujhē)" and " मैं (maiṁ)" in Hindi 
In November 2010, I joined a course of Hindi language through some continuous education centre. Although the tuition fee is very inexpensive (18 hours at a rate of USD70), I unfortunately met one of the worst teachers ever through my life in learning foreign languages: Every time when I posted a question related in Hindi language (such as grammar, sentence structure, vocabulary, or even simply pronunciation), she either did not want to answer, or she answered with very little patience. Hence, I have a question in this language.

Here comes two examples:
1) I do not know.
2) I do not speak Hindi language.
As we see, both sentences have a common nominative subject, i.e., I.

Nevertheless, from the class and through the Google translator, I learnt that the above two examples in Hindi become
1)　मुझे नहीं पता (Mujhē nahīṁ patā)
2)　मैं हिंदी भाषा बोलते नहीं है (Maiṁ hindī bhāṣā bōlatē nahīṁ hai)

I hereby would like to know what might be the difference(s) between " मुझे (mujhē)" and " मैं (maiṁ)". Why do we have different representation of subject pronoun? In what situation(s) do we use which pronoun, respectively?

Thank you very much for your kind attention and any answers or comments, and good day to all.

Answer

main is first person while mujhe actually is "to me"

strictly speaking in hindi grammar
main मैं is "Karta"(कर्त्ता) pratham vibhakti ekwachan
mujhe मुझे  is "karm"(कर्म) dwitiiya vibhakti ekwachan

dont worry even if it gets over ur head :D :)

"mujhe nahin pata" is accurate and can be conviniently used

But for ur convenience i would have translated these sentences as

मैं नहीं जानता/ जानती हूँ |
main nahin jaanta/jaanti hoon. (i dont know)

मैं हिंदी नहीं बोलता/बोलती हूँ |
main hindi nahin bolta/bolti hoon(i dont speak hindi)

one of critical error in second sentence is, whenever u start sentence with "main" मैं, it must end with hoon only and never with "hai, hain or ho"

another point to be added, the way i(region) speak, for negative sentence i just dont end it up with hoon(may be regional influence or something else but)

मैं नहीं जानता/ जानती

मैं हिंदी नहीं बोलता/बोलती

is a way to express in day to day life

adding further u must end positive sentence with hoon

ex.

मैं जानता/ जानती हूँ |(i know)

मैं हिंदी बोलता/बोलती हूँ |(i speak hindi)

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keeping it simple discard "hoon" omission rule, and make it a thumb rule that for present tense the form of "is" by which sentence must be completed is HOON हूँ 
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ex 
main jata hoon(male says "i go")
main khata hoon (m - i eat)
main khelti hoon (f - i play)
main ja rahi hoon (f- i m going)
we see present simple and present continuous taking hoon in ending whenever stated with "main"(its main and only main no other variant)

go through this i have acute shortage of time to write lengthy and analytical answer, do askif u r still in doubt

http://www.aclweb.org/anthology/W/W10/W10-3216.pdf

free word order in paninian framework(good one)
http://www.iiit.ac.in/~sangal/files/papers/acl93.pdf

Answer

मुझे नहीं पता = I don't know.  Is correct.
However in :  मैं हिंदी भाषा बोलते नहीं है.  It should be  BOLTI  not  BOLTE.  Bolti = Female,  Bolta = Male,  Bolte =  Plural and giving / receiving Respect.
It's very sad to know U had a bitter experience in learning Hindi.  I hope, it wont repeat.  Would U like to take a trial lesson.  Tell me, What is the best time for U to have it.  Would U be comfortable HERE (ON THIS SITE) OR ON sKYPE / ym etc.  My id is  ' mdchaskar '  U may add in your contacts and confirm accordingly.
Muze / mujhe = to me / to have = possessive.  and  Main = I = watashiva.  ok ?
Look forward to hearing from you.