What are the differences between the spirit of God and spirit of Man? First, I thought that the spirit Christians get from God is the same as the spirit of God. Recently, I heard a sermon that asserted they are different, but gave no more details.  Which verses describe this in the Bible?

When we get the Holy Spirit are we getting the spirit of God?
It helps to first know what the word "Spirit" actually means, which is more of a question for the Hermeneutics site.  I do see you've been asking questions about the meaning of the word there, so I'll include a brief bit here as introduction.

From http://www.pickle-publishing.com/papers/soul-and-spirit.htm


  In the New Testament the word for "spirit" is pneuma. Pneuma is
  translated the following ways: 
  
  
  ghost 2
  Ghost (with Holy) 90
  life 1
  spirit 151
  Spirit 137
  spiritual gift 1
  spiritually 1
  wind 1
  


The entire article is very long, but it does a good job of illustrating how a simple word can be interpreted different ways based on the context of the verse.

The word "Spirit" can carry the more religious/supernatural connotation associated with the word today, or it can simply mean "life" or "the breath of life". Quite commonly it is used as a literary form of personification:


  One brief thought before we look at the verses: There are some verses
  that seem to use "soul" and "spirit" in ways that harmonize with the
  common concept of the nature of man. How can this be? Is the Bible
  contradicting itself?
  
  Here is one suggestion: It was commonplace for the Bible writers to
  take parts of man’s being and personify them, give them attributes
  they did not in actuality possess. Perhaps sometimes they personified
  the "soul" and "spirit" as well.
  
  The most familiar example of a part of a person being personified is
  the heart. The heart, simply an organ that pumps blood, is said to
  have qualities that the mind does have, but that the heart definitely
  does not have. 
  
  Another example which is not so familiar is the personification of the
  kidneys, called the "reins" (Ps. 7:9; 16:7; 26:2; 73:21; Prov. 23:16;
  Jer. 11:20; 12:2; 17:10; 20:12; Rev. 2:23). The kidneys seem to have
  been made the seat of the affections and emotions. 
  
  Another example is the use of the words for "bowels" (Ps. 40:8
  (translated "heart"); Cant. 5:4; Is. 16:11; 63:15; Jer. 4:19; 31:20;
  Lam. 1:20; 2:11; Luke 1:78 (translated "tender"); 2 Cor 7:15
  (translated "inward affection"); Php. 2:1; Col. 3:12; Phm. 1:7, 20; 1
  Jn. 3:17). 
  
  In the light of these scriptures, the possibility that the Bible
  writers also occasionally personify the "soul" and "spirit" should be
  considered. In other words, the "soul" and the "spirit" may in some
  verses be given qualities that they do not in actuality possess.


The article shows the various verses in which the word "Spirit" is used.  I'll include just a few:


  "And the LORD God formed man of the dust of the ground, and breathed
  into his nostrils the breath (neshamah, pnoe) of life; and man became
  a living soul" (Gen. 2:7). [God put the "spirit" into Adam's nose.]
  
  "And, behold, I, even I, do bring a flood of waters upon the earth, to
  destroy all flesh, wherein is the breath (ruach, pneuma) of life, from
  under heaven; and every thing that is in the earth shall die" (Gen.
  6:17). [Animals have the "spirit" in them too.]
  
  "All the while my breath (neshamah, pnoe) is in me, and the spirit
  (ruach, pneuma) of God is in my nostrils" (Job 27:3). [The "spirit"
  lives in the nose.]
  
  "The Spirit (neshamah, pnoe) of God hath made me, and the breath
  (ruach, pneuma) of the Almighty hath given me life" (Job 33:4). [God's
  "spirit" gives us life.]


I'm assuming your question deals with verses like the last two shown above.  In the context, and general meaning. 

Job 27:3 has been translates several ways by scholars, in the various versions of the Bible.  Examples:


  GOD'S WORD® Translation (©1995) 'As long as there is one breath [left] in me and God's breath fills my nostrils,
  
  American King James Version All the while my breath is in me, and the spirit of God is in my nostrils;


Both of these clearly indicate that the Spirit of God is distinct from the breath of life (spirit of man).  It indicates that man has a spirit in him, given to him by God, not that the spirit in him is the same spirit that God possesses.

The spirit that God gives may be from God, but the connotation of the word "spirit" when it comes to man is simply life itself.  We each have our own life - our own spirit - our own ?breath that gives us life" that is certainly a gift from God, but is not the same as God's spirit.

It's also completely different than the Holy Spirit, which is the third person of God in the Trinity and, according to most traditions, indwells us at the point of, and remains in us after salvation.