In general, a CSI or CS tech actually visit a crime scene. They're the one's collecting (and submitting) evidence, and (typically) work at or very closely with a police or sheriff's office. While a forensic tech stays in a lab and actually tests the submitted evidence (in a sterile lab environment) according to their training.
This is where the State-by-State cases emerge. Some states train their Forensic Scientists to be a jack of all trades, they don't specialize in any one field and have some experience in numerous fields. Meanwhile some States separate their forensic scientists by specializations (Firearms, DNA, toxicology, Document Analysis, etc.) In these States CSI is just another specialization that falls under the umbrella of "forensic science."
Such States normally advertise the specialty with the position, however, such that you may see a sheriff's office advertising for a "Crime Scene Tech - Latent Prints." (Someone to go out to crime scenes and collect evidence/check for fingerprints.) Or a State Law Enforcement Agency asking for a "Forensic Tech - DNA Analysis." (Someone to stay in a lab and retrieve DNA from samples submitted to them.)