Difference between 兴趣 vs 意思?
In this context


没有兴趣 means "has no interest"  or "not interested" (not something you like to do or care about)
没有意思 means "meaningless", if something is meaningless to you, it is a given that you are not interested in it.


The implied meaning in the sentence "养宠物没有意思" is  "对我來説, 养宠物没有什么意思." (To me, keeping pet is meaningless)

Look at it the other way:

这玩意很有趣 This stuff is very interesting

我对这玩意很有兴趣 I am very interested in this stuff

这玩意很有意思 This stuff is very meaningful (therefore, your are interested in it)


  When to use one over the other?



When you say "没有兴趣" (I am not interested), you are expressing your personal preference about a topic, e.g. 'you may not like fishing, but you are not saying fishing is boring' 
When you say "没有意思" (meaningless to me), you are giving your low opinion on the topic, e.g. 'you really think fishing is waste of time'