Most forged dropouts are made from fairly mild steel in order to remain malleable for alignment purposes. Dropouts don't really affect ride quality in any way, but hanger design may influence what derailleur you will use on the dropout, what size cluster can be accommodated, etc. Slot design influences what type of drive train will work (e.g. vertical dropouts don't play well with fixed gear set-ups) and ease of mounting accessories like mudguards and racks. Rear-facing slots don't work well with mudguards, but facilitate wheel changes on tightly-coupled track frames. Vertical dropouts play quite nicely with mudguards. Eyelets facilitate mounting racks and mudguards, etc.

So, yes: dropouts do matter, but not because they affect how the frame feels while riding.