They were both government-run construction programs.
The Tennessee Valley Authority, established in 1933, sought to reduce problems in the Tennessee River valley caused by floods, deforestation, and soil erosion.. The TVA taught farmers improved methods, planted trees, and worked to construct dams. (That's the government-run construction part.) The TVA's dam projects generated much electrical power to the region also.
The Public Works Administration, also established in 1933, was devoted to public construction projects, such as bridges, roads, dams, airports, public meeting houses, and more. The PWA aimed not only at providing jobs but also at improving the infrastructure across the country. PWA projects spent over 7 billion dollars during its decade of operation, paid to construction firms contracted to hire workers and accomplish the various construction projects.
