We compare the fusion of giant lipid and block-copolymer vesicles on glass and poly(dimethylsiloxane) substrates. Both types of vesicles are similar in their ability to fuse to hydrophilic substrates and form patches with distinct heart or circular shapes. We use epifluorescence/confocal microscopy and atomic force microscopy on membrane patches to (i) characterize bilayer fluidity and patch-edge stability and (ii) follow the intermediate stages in the formation of continuous supported bilayers. Polymer membranes show much lower membrane fluidity and, unlike lipids, an inability of adjacent patches to fuse spontaneously into continuous membranes. We ascribe this effect to hydration repulsion forces acting between the patch edges, which can be diminished by increasing the sample temperature. We show that large areas of supported polymer membranes can be created by fusing giant vesicles on glass or poly(dimethylsiloxane) substrates and annealing their edges.