The first factor affecting the difference between these two works is the circumstances of the authors.
Hughes was born in 1902 and Meinke some 30 years later in 1932. Hughes was African-American, and though from an educated family, experienced several disruptions in his family life, and worked odd jobs before finally succeeding as a writer. He moved to New York's Harlem, where he lived for most of his life, and rarely accepted offers to teach at universities. Hughes never married.
Meinke, who is of white ancestry, was born in Flatbush, Brooklyn and grew up in New Jersey. He attended Hamilton, an elite liberal arts college, received a PhD from University of Minnesota, was a professor at Eckerd College in Florida from 1966 until his retirement in 1993, and is an active member of the academic creative writing community. He is married and has children.
"Mother To Son" by Langston Hughes reflects the situation of an African-American woman, living in poverty and enduring racial discrimination, struggling to make a good life for herself and her family. The tone is colloquial, using many distinctly African-American dialectical features. The mother urges the son to continue struggling to improve his life, never resting or relaxing, no matter how difficult the situation. She sees her struggles as a model for her son to emulate. The tone of the poem is one that prompts us to admire the mother's determination.
In comparison, "Advice to My Son" by J. Peter Meinke reflects a form of white privilege, taking for granted that neither narrator nor son really need to face a life of struggle for survival. Instead, the poem follows a "carpe diem" theme, suggesting that as life passes by quickly, the son should enjoy sensual pleasures such as beautiful flowers and fresh vegetables, and enjoy the luxury of relaxing with friends over wine. The son should, of course, marry a "pretty" girl (whether the girl has any intellectual or moral character doesn't seem to matter) and relax and enjoy life.