Here’s a summary of what is similar between the two formats:
Both are handmade processes & can have minor variations due to the vagaries of wood and inking.
Both are almost entirely made by me. (I sometimes have an assistant tear paper, carry prints to the drying rack, and help clean up.)
Both are made with archival inks, though not the same ink. Both are inspected for quality & packaged with acid free polybags & mats.
Woodcut is the original artwork and is considered a fine art print.
My woodcut prints are signed by me, and typically limited in edition size.
My woodcuts on paper are printed with enough pressure to slightly emboss the paper, which gives them a subtle visual effect that simply doesn’t translate well to photos and scans. The papers I use are archival quality and typically hand-torn instead of cut.
Woodcuts are much slower to print than screen prints. Wood wears down over time, and I use a soft carving wood, so I can only pull so many prints from the woodcut.
Screen prints (as I make them) are reproductions based on the original woodcuts.
The art is altered to better suit the wood veneer I print on, and also scaled down.
The ink applied to the wood does not get embossed.
Wood veneer is not technically considered archival, though it should be long lasting.
Screen prints are faster to produce and are unsigned. When the screen wears out, I can make new screens and continue printing as many as I want.