The sentences would mean the same, if you used the same pronoun in both sentences, like:
1) Wir mögen sie, weil sie uns mag.
2) Weil sie uns mag, mögen wir sie.
German word order rules require (with some conjunctions as exception) that in sub-clauses (e.g. one starting with "weil") the conjugated verb goes to the end, while in main clauses it goes to the second position.
In sentence 1) the positions of the words are easy to count: the main clause has "mögen" at position two: Wir mögen sie, while the sub-clause starting with "weil" has "mag" at the end: weil sie uns mag.
In sentence 2) you count the same as in 1) in the sub-clause, just that the sub-clause is at the beginning. But the main clause: mögen wir sie, seemingly contradicts the rule, as it starts with "mögen". Here you have to known, that the preceding sub-clause as a whole counts as position one, which make "mögen" in accordance with the rule for main clauses be at second position.