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Both of these verses use the same Greek word: pnuema, but in slightly different contexts. The first uses it as a name, the second as a description of essence. The Spirit of God (1 Cor. 12:3) is the man name used of the third person in the trinity whereas saying that God is a Spirit (Jn. 4:24) places the one God in the essential category of Spirit as opposed to within the category of material creatures or gods which must be worshiped within physical temples. Therefore, blueletterbible.org would categorize the first use under the "I." and the second use would fit definition "III."
An analogy would be the use of the word Son. It is one word but with several meanings and potential definitions. I may say: "I have a son" or "You are my son" or I might call you it as a name "Come over here Son!"