As I tried to listen to the real speaking environment, I can't really see the difference between those.
z.B:
- Ich habe eine nette Mutter.
- Ich helfe meiner netten Mutter.
Writing seems like the wrong medium to answer this question.
Wikipedia has a vowel chart that shows the difference between unstressed -e [ə] and -er [ɐ]: the latter is more open. This will give the vowel a quality somewhat similar to a. (In fact, you will frequently see intentional misspellings of Bruder, Alter, Dicker as Bruda, Alda, Digga.)
Here's a Youtube video. At around 1:58, there are some minimal pairs you can listen to. The difference is small but noticeable. Try pronouncing [ə] while consciously jerking your chin a bit downward, thereby opening your mouth. You will produce [ɐ].
meine > meiner
bitte > bitter
messe > Messer
denke > Denker
eine gute Freundin > ein guter Freund
etc.
The two are different phonemes to a German ear, so to a German they sound sufficiently different to be distinguished. However, the sounds are close. It has been pointed out that meine uses an e-shwa /ə/ while meiner uses what is often termed an a-shwa /ɐ/. This means that the latter will be a little deeper, a little more open, a little closer towards an a.
However, if these two sounds are not distinguished in your native language, it is hard for you to tell the difference in German. Other learners may have difficulty distinguishing i and ü or any other close pair that is non-phonemic in their native language. The only advice I can give you is to keep on practising, listening and trying. With time, you should be able to hear the distinction.
Some dialects render these phonemes (especially -er) in a different way that will certainly aide you if you happen to be in that area. In the Allgäu and probably other parts of the Swabian dialect area, the final r tends to be stressed or even syllabic, making the word sound more like meinrr. In Saxonian, the clichéd pronunciation of -er is -or. And a former colleague from the extreme North-East tended to pronounce -er as -ää. However, the terminal -e of meine will never stray too far from a standard /ə/.
That is not surprising. In theory, -er vs. -e, being different morphemes, could express important differences in meaning. In practice, the contexts where they alternate (e.g. "der große Wagen" vs. "ein großer Wagen" tend to be overspecified, i.e. other particles and constructions already define the grammatical role of these words unambiguously. As a result, the difference is not so much one meaning vs. another, but a correctly vs. incorrectly inflected expression of the same meaning.
It is not a coincidence that similar-sounding morphemes are used in such auxiliary roles; compare the huge difference between "er" and "sie"! Language constantly adapts itself to the needs of to speakers and hearers; the result is that elements which cannot deliver much disambiguation are shuffled towards lesser roles such as redundant inflection duty.
Feel free to improve:
ish huhbae inae nettae mootuh.
versus
ish helfae miner nettin mootuh.
or simply
minae
versus
miner
I think what he means by his question is why the pronoun "eine" is used in the first sentence and the "Genitiv" "einer" is used in the second.
The answer to this question is simple. You use "einer" if something is being done to something or something belongs to something.
"Eine" is being used in sentences where to something not specific is referenced.
"Meiner" however, is being used as a pronoun in the "genitiv" referring posession.