Your Granny Smith Apple tree has a bacterial disease called fire blight.   This disease is very common on apples, crabapples and pears in our area.  Granny Smith apple is a variety that is highly susceptible to fire blight.   Some more photos below.
Details and control options:
http://www.ext.colostate.edu/pubs/garden/02907.html

Thank you for your advice!  the information sheet you sent says to cut back the diseased branches - would you recommend doing that now, or waiting until winter when the tree is dormant?  Thanks!

Plant Pathologists are split on when to do that pruning.    Some say prune back as soon as possible after symptoms, as long as you cut several inches below cankers and sterilize pruner blades after every cut.    Others say wait to prune until the dormant season (Feb-Mar), when the bacteria are not active and there's less chance of spreading infection with pruning tools.
I'd go with doing it now if the visual symptoms are upsetting, you feel confident you can prune as described in fact sheet and will sterilize blades between cuts.