This is how I started. A DA's office that utilizes horizontal prosecution will be made up of separate bureaus that transfer cases between them.
A felony arrest occurs and the case is immediately assigned to the grand jury bureau. An ADA in the bureau analyzes the case and decides if it will remain a felony or proceed as a misdemeanor. If it stays a felony, the ADA investigates the case, requests any laboratory testing, interviews the witnesses, sends out subpoenas and search warrants, and finally presents the case to the grand jury. After the grand jury indicts the case, the case is forwarded to a different ADA in a trial bureau who handles the case until it is resolved through plea or trial.
My current office uses vertical prosecution. As opposed to horizontal, one ADA handles the case from the grand jury investigation through trial. There is no transfer of cases or a grand jury bureau dedicated to investigating cases.
The main advantage of vertical prosecution is that one ADA handles the case from beginning to end. That ADA is then intimately familiar with all the relevant facts and has (hopefully) fully investigated the case from the beginning. The ADA in a vertical prosecution office presumably has done all of the case investigation before grand jury because that ADA knows they will always have the case.
Another advantages of vertical prosecution are that the victims and witnesses have continuity in their prosecutor and don't have to tell the same story many times.
The benefit of a horizontal prosecution office is that it allows an ADA to focus on their specialty. In the grand jury bureau, the only concern is investigating and indicting cases. A trial ADA is only concerned with trying cases. Neither ADA is required to be in every place at once. An ADA in a vertical prosecution must handle felony hearings in city court, investigate and present cases to the grand jury, try cases, and appear in every courtroom when one of their cases is on the court calendar.
Another advantage is that horizontal prosecution provides a natural training progression. An ADA learns the laws, practice of grand jury, and necessary investigative steps required to prepare a case for indictment. That ADA will spend a year or so doing this before moving on to the trial bureau. A trial ADA starts with low level felony cases (like DWIs or guns) and moves toward serious victim crimes.
In a horizontal office, you are always relying on another person's work. As in any job, some ADA's are more diligent than others. When you receive a file that is organized and fully investigated the life of a trial ADA is easier. When you receive a file with paperwork everywhere and only the minimal amount of work done, headaches ensue. In a vertical office, you have no one to blame for your work.
Even in horizontal prosecution offices, specialized bureaus like sex crimes, domestic violence, and white collar crimes are handled in a vertical manner by ADA's specially trained for those cases.
Which do I like better? Like I said the grass is always greener. When I left a horizontal office, I was excited to handle a case from beginning to end. Now that I'm in a vertical office, my energies are drained when I'm pulled in city court, supreme court, grand jury, and trial all at the same time and some days I miss the horizontal prosecution of cases.