The first sentence is an example of the use of the present perfect continuous/progressive. That is present tense + perfect aspect + progressive aspect.
The second sentence is an example of the present perfect - present tense + perfect aspect.
From this we see that grammatically the only difference between them is the progressive aspect. How does this affect the meaning? Well, the progressive aspect adds the meaning of time limitation in some way.
So, the sentences you provided can be explained as follows:
I have been living in Paris. This tells us that the speaker believes that this is or was a temporary situation (time limited) - perhaps it is a student who normally lives in London but recently lived and studied in Paris for a period of time. We can imagine a situation where two friends meet in the street and one says to the other -
'I haven't seen you for a long time. Where have you been?
The other would then naturally reply -
'I've been living in Paris.'
It would be wrong to reply 'I have lived in Paris.'
I have lived in Paris. This is a simple fact and tells us that the speaker at some time before now lived in Paris. Here the length of time the speaker lived there is not considered important - It is just a statement of fact.
When we add for/since to these sentences:
'I have been living in Paris for the last 6 months/since April.'
We understand when our student began living in Paris and that this situation has either finished recently or continues but the speaker believes it will finish at some point in the future e.g. when she/he finishes their course.
'I have lived in Paris for 6 months/since April.'
Now we understand that the speaker lives in Paris and started to live there 6 months ago. The choice of present perfect simple here tells us that the speaker is making a statement of fact. This could be either because the speaker believes this is a permanent situation (he/she moved to Paris), or for some other reason. For example, our student who is temporarily living in Paris might have been stopped by a policeman. The policeman is only interested in present facts so when he asks how long our student has been in Paris our student could well choose to answer 'I have lived here (in Paris) since April.' even though the situation is temporary in the student's mind.
So to sum up:
If the situation is time limited, we usually use present perfect continuous/progressive.
If the situation is either not time limited or the speaker decides to make just a factual statement for some reason, we choose the present perfect simple.
Sorry, I seem to have spent a lot of time explaining this.
- For Teachers