Each turret type has its own advantages and disadvantages.
BMT turret is often installed on turning centre with live tooling. The drive is hidden inside the turret and does not hinder machining, while the radial and axial live tool holders allow the use of both main spindle and sub-spindle.
BOT turret is a good choice for a common 2-axis lathe. Radial external turning tools are installed directly into the turret slots without the use of tool holders. This turret also allows operations with a sub-spindle.
VDI turret can not be used with a sub-spindle, but allows application of live tools. There is a wide range of VDI tool holders. They are cheap and can be easily replaced. This is convenient for retooling.