There are plenty of comparisons and also differences between these two poems. Both are clearly Romantic works in the way that they are meditations on nature and nature is described, and how it acts as an inpiration to our imagination and how nature compares to the urban squalor enjoyed by so many in this period of Industrialisation. Also, both poems directly refer to the way in which adulthood and childhood are related in the memory in relation to nature.
One of the key differences, however, is the way in which childhood is regarded. In "Tintern Abby," Wordsworth, who was raised in the countryside, talks about the way in which his childhood was a time of intense connection with nature. In various poems he deilberately goes back to his childhood as a source of inspiration and also to soothe his feelings. Coleridge did not enjoy such an idyllic childhood, and was raised in London, "pent 'mid cloisers dim," and he in this poem finds Wordsworth's easy link with childhood and nature problematic. Coleridge during his childhood says he "saw naught lovely but the stars and sky" and this clearly was a struggle for his childhood.
Coleridge in "Frost at Midnight" therefore determines that his child will enjoy the kind of innocent and rustic upbringing that Wordsworth enjoyed, "by lakes and sandy shores, beneath the crags / Of ancient mountain, and beneath the clouds." One of the key differences between these two poems is therefore the way in which Coleridge presents the link between childhood and nature as being something that cannot be taken for granted, His own lack of this connection is something that demonstrates this key difference.