Both advices have merit, however it entirely depends on the context.
The perceived sincerity of apology delivered to someone who is receptive to an apology can indeed be diminished by accompanying it with an excuse.
On the other hand, an apology delivered to someone who is categorically unreceptive to any apology can only be entirely futile. Since there is nothing to gain yet possibly something to lose, the rational course of action would be to not apologize.
Edit: since the latter example seems to be the controversial one, here is a popular scenario in which a significant number of individuals can always be characterized as being "categorically unreceptive to an apology": partisan politics.
The perceived sincerity of apology delivered to someone who is receptive to an apology can indeed be diminished by accompanying it with an excuse.
On the other hand, an apology delivered to someone who is categorically unreceptive to any apology can only be entirely futile. Since there is nothing to gain yet possibly something to lose, the rational course of action would be to not apologize.
Edit: since the latter example seems to be the controversial one, here is a popular scenario in which a significant number of individuals can always be characterized as being "categorically unreceptive to an apology": partisan politics.