Why don't banks calculate interest in CD using continuous compounding (which is higher than compounding quarterly)? While in the case of a mortgage they use continuous compounding.
Don't know what you're talking about. Just checked the Wells Fargo web site - they compound the CD interest daily.
Differences are entirely due to convention, and conventions may differ between banks or even between products at the same bank. This is why the "APR" exists, with a precise legal definition allowing one to compare rates. The equivalent continuously compounded rate can easily be found for any discretely compounding rate, and vice versa.