This is a question that I frequently get from clients & artists who are looking to record. With clients budgets being more modest and "in studio" recording times decreasing, artists are having to take a harder look at all session expenses. Engineers & Producers are part of the equation and these two jobs are extremely important to the recording process. This is my attempt simply to educate, not to imply that "this is how every session works in every situation".
Generally, an engineer and a producer are listening to two different aspects of the same piece of music being recorded. In a typical session there are two engineers a 1st & a 2nd who handle & run all of the equipment needed for capturing sound. A 1st engineer is in charge of the equipment & the quality of sound that is coming through the mixing console/board going to "tape" (Pro Tools). Typically they'll pick out some preferred microphones, outboard gear, and vocal chain that he or she likes. A 2nd is the 1st's assistant and does whatever the 1st doesn't have enough hands to do in the moment. The 2nd engineer usually set's up & tears down the session, patches the mics, and is there for any session needs that arises unexpectedly. Typically, as a "stereotype" engineers are in love with different tones, sounds, frequencies, outboard gear (or any devise that does sound manipulation), compression, eq's, computers, file management, Pro Tools recording software, etc. We lovingly refer to them as the "button pushers" who just love to fiddle with knobs and switched. However, they are truly "engineers" in the literal sense, only these engineers design, manipulate, and build audio projects.
With that said, I have found that people are the most confused about what a music producer actually does. A producer, from an artists perspective, is your best friend and advocate in a recording session. Producers work with the artist to develop & enhance the range & scope of the artists composition. Producers focus primarily on the music being recorded & managing the funds required to do so. We handle the musical arrangement, song tempo, lyrics & chords, number charting, last minute rewrites, musical coaching & performance, hiring musicians, developing budgets & keeping the project on schedule, coordinating session times, all personnel payments, union cards, invoices, overseeing the entire session, to name a few. A producer will be able to push you passed your perceived limitations (that you have put on yourself) to get the best performances out of you. A good producer will understand the artist and know (like no one else can) what they desire their music to sound like. A great producer must be able to honor the artists vision (while at the same time) keeping what sells & is marketable in mind. When you have a producer on your side you'll have a fun experience recording, be as efficient as possible (to keep your budget low), and perform to the best of your abilities.