Lay not up for yourselves treasures upon earth, where moth and rust doth corrupt, and where thieves break through and steal: But lay up for yourselves treasures in heaven, where neither moth nor rust doth corrupt, and where thieves do not break through nor steal: For where your treasure is, there will your heart be also. (Matthew 6:19)
Both Stoicism and Buddhism (accurately) see the negation of attachments as a solution to suffering. Therefore, if one removes the attachment, one lessens the suffering when those things are destroyed.
Although Christianity also (accurately) sees this, this isn’t the primary motivation for removing attachment to things in Christianity. Consider that Stoicism tends to go further, rejecting attachment not just to things (i.e., hedonism or materialism) but to people and relationships. Buddhism also has this element.
Differently, in Christianity the primary motivation for rejecting attachment to materialism or hedonism comes not from considerations of suffering but from an alignment of one’s will with the will of the Good, i.e., a rejection of egoism.
Stoicism is the opposite – a rejection of attachment stems from egoistic considerations.
The basic idea in Christianity is that one aligns one’s interests with the interests of the Good. Yet, almost paradoxically, because ‘happiness’ or ‘meaning’ is rarely found by focusing on one’s self, this re-alignment tends to lead to happiness. I.e., happiness or meaning is found by focusing outside oneself (on family, on a cause, on the good, and so on), and valuing these things.
To put the basic idea in more Christian terms, by losing oneself, one gains oneself. Stoicism (and Buddhism, to an extent) misses this dynamic in motivation, and so settles into a kind of nihilism, where any ‘goodness’ is suspect.