The Stevens 44 1/2 is a falling block action, meaning the breech block moves up and down within the frame, while the 44 breech block pivots on a central pin. Here's a useful illustration from rimfirecentral.com:
With a few exceptions, parts are not interchangeable between the different models. Stocks made for a 44 1/2 will fit a 44, though since the 44 1/2 is thicker you'll need to trim excess wood.