What is the difference between spirit and soul? Is the word soul used for only human beings? For instance,
He [Descartes] thought the brain worked as a center for the spirits of the 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:
“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:
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.
The Catholic apologist Frank Sheed, in his book Theology for Beginners, explained soul as being the animating principle in a living thing. Humans have souls. Cats have souls. Even plants have a soul. The soul of a human, as a sentient being, is also a spirit.
This may be the sense of the quotation from Descartes, who at one point in his life studied at a Jesuit college. The human body is the seat of the soul while the human brain is the seat of the sentient spirit.
The earliest manuscripts of the Bible do not, of course, contain any of the words soul, spirit, or heart, as they were not written in English.
Few Bible scholars would endorse the view of Frank Sheed that plants possess souls.
The tripartite view of man adopts the model that man has
(1) a body, the 'outer man', with which the doctor (or medical examiner!) is familiar;
(2) the 'inner man', with which the psychologist and social worker are familiar (this is often called the soul in this model). This 'contains' the thinking (cognitive), the emotive (often called the 'heart'), and the deciding (the 'will') 'centres';
(3) the'innermost man', that which relates (or would do if energised) with God. This is always labelled the spirit. Hillman (see above) echoes this view, albeit rather faintly.
Sentient creatures would thus have a soul (dogs can be quite smart, show rudimentary to quite complex emotions, and can refuse to obey). Plants would not. Only man seeks to relate to a non-corporeal higher power: he alone is a spiritual being. The Genesis description of creation uses the word bara (create) three times, in line with these three 'orders' within creation.
On this model, the spirit-soul complex persists after physical death. See, for example, "The Holy Spirit and You" by Dennis and Rita Bennett.
The bipartite view of man condenses 'inner' and 'innermost' man.
The Bible seems to use either model according to which better illustrates the point being made in that particular passage. Neither is a totally accurate description of actuality.
See "Aglow With The Spirit" by Robert C. Frost
As has been pointed out, when we start using the English terms spirit and soul to translate ruach/pneuma and nephesh/psuche from the Hebrew/Greek originals, the polysemes (different individual senses) of the English replacements are many, and do not correspond exactly with those of the originals. And heart seems to be used almost interchangeably with both soul and spirit.
Spirit is the knowledge capability which God inflated humans with, from his own ultimate divine knowledge. Characteristics distinguishing humans from other animals are all related to the Spirit.
Soul is the part that ignites life into all livings. When taken away life stops and fail to continue.