The central difference between these two poems lies in the way that "The Garden" is an evocation of man's garden and domesticated plants, flowers and trees as a place of peace and tranquility that provides balm from our busy lives. By contrast, "The Mower Against the Garden" deplores the way that wild, untamed nature has been "civilised" and boxed in by man into a garden, and sees the garden as profoundly unnatural.
In "The Garden," the speaker is clearly incredibly grateful for the peace and quiet that he is able to find in the garden:
Fair Quiet, have I found thee here,
And Innocence, thy sister dear!
Mistaken long, I sought you then
In busy companies of men
A division is thus drawn between the world of men and their "busy companies" and the quiet and innocence that can be seen in the garden itself, which the speaker describes to be some form of retreat from the tiring world.
By contrast, note how the speaker of "The Mower Against the Garden" talks disparagingly of the garden, describing it as a "dead and standing pool of air," and saying that man created "Forbidden mixtures" in the garden. For him, the purest evocation of nature is seen in the untrammelled wilderness of the field:
While the sweet fields do lie forgot,
Where willing Nature does to all dispense
A wild and fragrant innocence ;
And fauns and fairies do the meadows till
More by their presence than their skill.
In contrast with the "enforced" nature of the garden, true innocence is to be seen in the "wild and fragrant innocence" of places where man's touch has not domesticated it and subverted it.