The only comparison I can see is that in "A Red, Red Rose" speaks of the newness of life in the season of early summer. It is a time of promise, and the tone of the poem is filled with promise. He says that he will love her for all time. He is leaving her, but promises to return, even if he has to travel a thousand miles to do so.
In comparison, "To A Mouse" has a very different tone: the opposite, in fact. He is sympathetic to the mouse's future that seems to hold little promise for survival, and notes that he feels much the same way. He feels responsible for what he has done, and as he continues to speak about his own difficulties, perhaps he is inferring that he has caused his own heartaches as well.
And fare thee weel, my only Luve,
And fare thee weel awhile!
And I will come again, my Luve,
Tho' it ware ten thousand mile.
That wee-bit heap o' leaves an' stibble,
Has cost thee monie a weary nibble!
Now thou's turn'd out, for a' thy trouble,
But house or hald.
To thole the Winter's sleety dribble,
An' cranreuch cauld!
In that the mouse thought she was ready for the harsh conditions of the cold months and had worked to provide for herself, perhaps he, too, has done the same, but misfortune has fallen on him as it has on her: he says they both face hard times and struggles, and that they can neither of them be assured that there will be any reward at the end of these struggles.
He believes that while her problems may only be momentary, when he looks back at his life and believes the problems will reach far into his future.
Still, thou art blest, compar'd wi' me!
The present only toucheth thee:
But Och! I backward cast my e'e,
On prospects drear!
An' forward, tho' I canna see,
I guess an' fear!
This contradicts the sense in "To a Red, Red Rose," that his love will last far into the future.