I think one of the best ways to think about the similarity in terms of theme and message in these poems is to try and relate them to a very common yet difficult to describe sensation that we are all subject to as humans. Have you ever felt there is more out there than you can somehow sense or put your finger on? Do you ever feel that your life amounts to more than it appears to do in your everyday, humdrum existence? Do you wish to connect to that sense of something divine or explore to search it out?
Both of the speakers in these poems seem to fit into this category: both are actively seeking to transcend their human limitations to explore that sense of the divine. In "A Noiseless, Patient Spider," it is the sight of a spider endlessly throwing out filaments to connect a web together that leads the speaker to explore how this image is true in his own life and for his own soul which throwns out filaments:
seeking the spheres, to connect them;
Till the bridge you will need, be form'd--till the ductile anchor hold;
Till the goassamer thread you fling, catch somewhere, O my Soul.
As E. M. Forster might have said, there is the intrinsic need to "only connect," and Whitman imagines that this is the soul's ceaseless activity as it tries to "connect" the spheres and link the soul to some kind of tangible sense of the divine.
In "Facing West from California's Shores," the search is embodied in a literal quest that the speaker has engaged him that has taken him round the entire globe:
the circle almost circled;
For, starting westward from Hindustan, from the vales of Kashmere,
From Asia—from the north—from the God, the sage, and the hero,
From the south—from the flowery peninsulas, and the spice islands;
Long having wander'd since—round the earth having wander'd,
Now I face home again—very pleas'd and joyous;
(But where is what I started for, so long ago?)
However, what is key to note is that in spite of his expectations, this ceaseless globetrotting has not brought him the kind of wisdom or knowledge that he thought it might bring. Even though he has almost "circled" the circle of the globe, he still is forced to recognise that, to quote U2, he still hasn't found what he's looking for. Just as the soul in the first poem endlessly throws out filaments to gain some conception of the divine, so the person in this poem seems to need to compulsively travel or voyage, literally or psychologically, in order to find what he is looking for. The quest of the human soul perhaps only ends when we die as humans.