Both classes of makruh acts are considered reprehensible and should be avoided (and doing so is a blessed act which would be rewarded); the main difference between a makruh tanzih act and a makruh tahrim act is whether performing them would be actually be considered sinful. On a scale of severity, they would look like this:
Mubah < Makruh Tanzih < Makruh Tahrim < Haram
Performing a makruh tanzih act is strongly discouraged, but would not necessarily warrant punishment in the hereafter (although to my understanding, many scholars claim that consistently and repeatedly performing makruh tanzih acts would be considered sinful, even if the single acts themselves would not). If something is makruh tahrim, on the other hand, it would be considered sinful and punished appropriately in the hereafter, but they are not so severe as to take one out of the fold of Islam (as would, for example, rejecting a haram ruling).