Both can cause urethritis and the feared complication, pelvic inflammatory disease. Urethritis, but not cystitis, will give you discharge.
N. gonorrhea: Gram - diplococci.
C. trachomatis: you might see a bunch of neutrophils, but no organisms. That's because Chlamydia are obligate intracellular organisms.
For treatment, a 3rd generation cephalosporin like ceftriaxone does the trick for Neisseria species. You can treat Chlamydia with either a macrolide like azithromycin or a tetracycline like doxycycline. But you don't want to give doxy to a pregnant patient. Often, you'll treat patients with both antibiotics because co-infection is so common.