Reciprocating saws and a Sawzall are the exact same tool. Sawzall is the trademark name that Milwaukee has for its range of reciprocating saws.
So Sawzall is a trademarked way of saying reciprocating saw, think of it like a Dyson being a trademarked vacuum cleaner. The name, however, has leaked into general usage for all reciprocating saws, just like a lot of people will call a vacuum cleaner hoover.
To add extra complication into the matter, reciprocating saws are often also called sabre saws. If you are in the market for a new reciprocating saw, make sure to read our article on the best sabre saws.
A Sawzall or Reciprocating Saw or Sabre saw has a lot of functions. As the trademarked name suggests, it will saw pretty much anything. The uses can go from demolition to gardening; it really is a multi-function tool.