What's the difference between sleep and standby mode? Well, for digital cameras, it's not very clearly defined. Indeed, the terms are used by many camera manufacturers (and camera users) interchangeably.
In laptops, there is no formal difference as defined by the ACPI standard (Advanced Configuration and Power Interface.) They are both classified under G1, the first state beneath working (G0). This is in turn subcategorized to 4 more states. What is commonly referred to as “sleep”, “standby” or “suspend” modes is S3. Turning a camera completely off is G3, “mechanical off”. Digital cameras can also follow these designations, as they largely consist of a computer than makes the camera function.
What formally defines this stage is that in addition to the CPU and non-essential functions being powered down, for a camera mostly just the LCD screen, and everything is suspended to RAM memory. This allows for the camera to almost instantaneously start back up again, as power is still running through the CPU, and the lens is still extended, yet still save power.
On many cameras, the shutter will also close during sleep or standby mode. This is to prevent the lens from being damaged, as many people will forget that the camera is on while not immediately shooting, holding it at their sides or at their waist, etc. This will add a couple of milliseconds to the time it takes to get from sleep mode to ready to shoot, but for most users, it's worth it.