Should I restore my old mt-bike for winter commuting (and for wife to ride with me) or throw it away?
Two questions here - restore an old bike and what bike for your wife. 

Lets deal with the easy one - an old Specialised will be a better bike than a department store one. If you need much more gear to be replaced, consider looking for a donor.  Add another $100 to you list for things that might need doing - like new cluster and chain rings, brake pads etc. Strip and repack wheel bearings and headset and you are probably good to go. From the sound of it though, a second hand bike might be a better option, as the deterioration that requires those parts replaced almost certainly means other parts are on their last legs. Cannot tell for sure without seeing it.  

Next one - What bike should your wife ride - that depends on your relationship. 
I see this all the time- Fit, active strong guy out on the trials riding a multi-thousand dollar ultra efficient lightweight bike miles ahead of his partner who clearly does not live breath and dream cycling like he does and is nowhere near as fit. She is struggling on an old beater her partner would not be seen dead riding, miles behind, exhausted and close to tears.......

If you don't like your wife, and/or want to be sent our riding on your own with her blessing, give her the beater. 

If you like your wife, and/or would like her to like you, enjoy riding and spend quality cycling time with you, give her the nicest bike and ride the beater. If she gets to like riding, you can buy her a nice bike that fits her properly. Play you cards right, you can buy her a better bike than yours, and when you go out solo - ride that...... 

(Presumption I have made is that the bikes fit both riders.  My wife and me fit the same frame size. I have two seat posts and seats for "her" bike - one set up for for her, one for me.....)