I've been investing time lately in improving my reading, and I have been mixing it up between sheet music and chord sheets. It's slow going but i'm improving, which is the main thing. Last night i got to thinking about which one gives me more bang for my buck in musical situations. I know the advice here will be "learn both", which I am doing.. But, in my head, it would seem that a more common scenario would be that I would be handed a chord sheet rather than a piece of sheet music that I need to play note for note. I dont plan on being a pro, or a session player, but I do plan on finding other musicians to play and jam with, and maybe some open mic jazz nights in the future. I am finding much more enjoyment and value from playing to chord sheets as it's really helping with my (lack of) improvisation skills.
So, which is the more common scenario you find yourself in? Being handed a sheet that you have to play note for note, or being handed a chord sheet and you deciding what to play? Would you rather be "ok" at sight-reading sheet and a demon with a chord sheet, or vice versa? And yes, i know, demon at both would be the ideal situation
If you play jazz combo from the Real Book you will generally play from a chord lead sheet, unless you are expected to play the melody. Then it's good to be able to read treble cleff. There are a few tunes in the Real Book that have a written out bass line. If someone brings in an original composition, they may expect you to read a bass line they have written.
You can buy a bass clef version of the Real Book but I recommend the treble clef YMMV.
An authentic jazz musician is expected to know a lot of standards from memory. Most people will tell you that learning the melody is a key step. Also you will normally be expected to solo freely over the changes. I can solo fine, but playing head charts is a skill I never mastered, so I don't consider myself an authentic jazz musician. But as long as I have a lead sheet or music in front of me, I am entirely comfortable.
If you are playing in big band, you will need to read written notation and chord notation. A majority of what you will do, in my experience, is read chord notation, but there are parts where you need to be able to read and play exactly what is written. The bass lines for some compositions are written out entirely note for note, some are written out with chord and rhythmic notation, so you know where to play the hits with the drums or the rest of the ensemble. Also, if you play big band for any length of time you will probably come across a few soli sections. Solis are a written out lines that you play with other people in the band (essentially group solos). Solis are often fast and technically challenging, and they should be played precisely.
IMHO interpreting chord notation is really what jazz is about. This is a very deep subject that relies on your ability to learn and apply music theory.
You need to be proficient in reading written music as well, but in my experience most jazz musicians are not monster site readers.
I guess what I am saying is you do need to do both, but I suggest spending more time on studying theory, interpreting chord notation, and memorizing head charts.
For the record I had no problem memorizing 4 or more hours of music for cover bands, and I was singing back ground vocals in addition to playing bass. But I tend to get lost in the more complex harmonic progressions of jazz standards. The challenge of memorizing one jazz standard is sort of equivalent to memorizing a whole set of pop music. Also, since my job did not require me to memorize jazz standards, I did not do it.