Spin-off: Company P creates Compay S by spinning off one of its division/business line and gives each of P's shareholder his/her portion of shares of S. So P's shareholders owns P's and S's shares after spin-off. One company becomes two with same base of shareholders, assuming all other things hold unchanged.
Split-off: Company P creates Company S and some of P's shareholders get S's share and give up some or all of their P's shares for exchange. The key here is they have to give up portion or all of their old shares for acquiring new shares.

Although the two restructuring models are similar. The value of the shares of P and S after restructuring could be differ a lot.