To delve into the historical side, Jesus's Jewish listeners, "this age" would have likely referred to the current era, marked by the Law and the Prophets, while the "age to come" represented the future Messianic era when God's kingdom would be fully established.
On the other hand, the Christian concept of the "age to come" has undergone evolution over time. Initially, Christians looked forward to the realization of God's kingdom. However, this anticipation has been redefined through the lens of Jesus's life, death, and resurrection. In Christian theology, the distinction between "this age" and the "age to come" is perceived as less distinct, marking the commencement of God's kingdom with Jesus's ministry and continuing through the work of the Holy Spirit in the Church.
Ellicott's Commentary for English Readers
Neither in this world, neither in the world to come.—The distinction
was hardly the same for our Lord’s Jewish listeners as it has come to
be with us. For them “this world”—better, perhaps, this age—was the
time before the coming of the Christ; “the age to come” was that which
was to follow it. (Comp. Hebrews 6:5, Luke 18:30.)
Pulpit Commentary
"The age to come" (העולם הבא) included all that followed the coming of
Messiah. Sometimes it was restricted to, or practically identified
with, the reign of Messiah upon earth, but usually it included much
more - eternity as well as time (see especially Weber, 'System,' pp.
354, 355; and cf. Schurer, II. 2:177). It is in its widest sense that
our Lord here uses it - contrasting the present order of things with
that which will be the final result of his coming, his thoughts
travelling far beyond the present course of this world to that which
is to be hereafter. Matthew 12:32
Conclusion: While initially important to the Jewish people as the beginning of a new age, Christianity later blurred the distinction between “this age” and “age to come,” seeing the beginning of God’s kingdom in Jesus’ ministry and the continuing work of the holy spirit.