The “flesh” and the “old man” are synonymous.

During his ministry, Jesus told his apostles that in order to follow him, one needed to “deny oneself” and “take up his cross” (Matthew 16:24). He further tells them that any who lose their lives for his sake shall find it (Matthew 16:25). 

Jesus devoted his complete mind, heart, time, resources and even life itself to serving God. When questioned about what the greatest commandment in God’s law was, Jesus replied that it was to love God with all one’s heart, soul and mind (Matthew 22:37). Such a complete dedication, or consecration, to God, comes at the expense of other pursuits or priorities in one’s life. If one is spending time in dedicated service to God, there will, of necessity, be less time for pursuits that many in the world find important or rewarding. Jesus referred to this concept when he explained that no one can serve two masters. If one tries, one will end up loving one more than the other, and resenting and hating the one that is taking time away from the activity that one really desires to do (Matthew 6:24). 

Setting aside worldly priorities and pursuits requires commitment and dedication. The Apostle Paul referred to this dedication as “the one thing he did” and that the priorities of the world were things he “left behind” (Philippians 3:13-14). Paul referred to this same concept, but using different phrasing, in Romans 6:6. When one consecrates one’s life to God, the changes that begin in one’s life are profound. As the mind, character and heart begin to reflect Jesus’ attitude of devotion to God, the person that one was is left more and more behind (an “old man” in Paul’s metaphor) as opposed to the new, emerging mind, character and heart that, with God’s help, one is building. As one begins to devote more time and thoughts and energy toward God’s service, one willingly gives up spending time or thoughts or energies on other things. In Paul’s metaphor in Romans 6:6, he likens this to “crucifying the old man on the cross.” This phrasing reflects that used by Jesus when he taught that his followers needed to “take up their crosses and follow him.” In your question, the flesh and the old man are synonymous.

Additional Resource: 
Christian Questions Podcast
Episode #1324: “What Is the New Creation?”
The profound results of God creating something never seen before
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