Different terms were introduced for the same concept when the Sudoku community developed their solving techniques. The term conjugate pair is mainly used in coloring techniques. The term strong link refers to the link between 2 candidates, but does not include the candidates themselves, whereas the term conjugate pair clearly refers to the candidates. Furthermore, a strong link can also exist between the last 2 candidates in a single cell, but it is not customary to call these 2 candidates a conjugate pair.
There is also a difference between a conjugate pair and a connected pair. In a conjugate pair, only one of the two candidates in the house can be true. In a connected pair, at least one of the candidates must be true, but it is also possible that both of them are true. The only certainty we have about a connected pair is that they cannot both be false at the same time.