Difference between "spirit" and "soul"
While you'll likely find no definitive answer, under general usage:
Spirit refers to a non-physical part of a sentient being that is the source of their emotions.
Soul is generally spoken about with greater religious implications, and can be considered an immortal, metaphysical aspect of a sentient being which is able to transcend the body and mind.
The words spirit and soul have many meanings and, as such, discussing their differences without particular context is difficult. I will, thus, answer to your more specific questions first:
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“Is word soul using for only human being in English?” — No. For example, the following is the Soul of a Unicorn:
Moreover, the New Oxford American Dictionary defines soul as “the spiritual or immaterial part of a human being or animal” (emphasis is mine).
Regarding your quote about Descartes: “spirits of the soul” is not an established expression, and I don't really see how it is different from, e.g., “[he] thought the brain worked as a center for the soul”.
On the overall differences between soul and spirit, I would say they are mostly interchangeable in their main meaning. For example, the New Oxford American Dictionary lists:
spirit (noun)
1. the nonphysical part of a person that is the seat of emotions and character; the soul
The Wikipedia article on soul actually discusses the slight differences in meaning, though it's far from clear-cut:
Although the terms soul and spirit are sometimes used interchangeably, soul may denote a more worldly and less transcendent aspect of a person. According to psychologist James Hillman, soul has an affinity for negative thoughts and images, whereas spirit seeks to rise above the entanglements of life and death. The words soul and psyche can also be treated synonymously, although psyche has more physical connotations, whereas soul is connected more closely to spirituality and religion.