- He left, although I begged him not to.
- He left, but I begged him not to.
- He left, and I begged him not to.
These three sentence do not have the same meaning, and the first is far more natural than the other two.
The first means "He left, in spite of my begging him not to." In short the begging did not have its expected or desired effect.
The second suggests that the begging was somehow not expected to occur, but did. A better sentence using a similar construction would be:
He left, but I wouldn't go with him.
In this sentence the leaving happens first, and the decision of the speaker not to go with "him" seems to happen after. There may be a suggestion that the speaker was expected to go. Here "although" could be substituted for "with" and the meaning would not change much.
The third sentence says that other person left, and afterwards the speaker begged him not to. This makes little sense. Better would be:
He left, and I begged him to come back.
In this kind of construction "and" usually indicates a simple sequence, and can often be replaced by "then" or "and then".