Typically, a turbo with physical small turbine and compressor wheels will have low inertia and therefore will accelerate faster for a given exhaust flow. However, because of their small size, the amount of the boost they can produce will be limited at higher engine speeds. Conversely, a turbo with physically larger turbine and compressor wheels will have higher inertia and will take longer to get up to speed (spool up), with increased lag. However, the benefit is that they will produce more boost.
Typically, the smaller turbo will produce a smooth power delivery but run out of puff early on, similar to a supercharger, while the larger turbo will give a fair kick when it does come on boost and hold that pressure until later in the rev engine. However, this can make the car difficult to drive, with an on/off nature.
Various methods of minimising turbo lag have been attempted over the years, to get a combination of boost pressures and hence power, along with throttle response and derivability. Mazda used sequential turbo on the last of the RX-7 models, with a small turbo spinning at low to midrange engine speeds and then a bigger version taking over at higher speeds.
The current crop of World Rally cars probably have the most effective system, known as anti-lag (ALS). Details vary from manufacturer to manufacturer but they all work on the same basic principle. A small amount of compressed charge air bypasses the throttle when it’s closed. The ignition spark is retarded at the same time and unburnt fuel is dumped directly into the exhaust turbine housing, where it mixes with the air and is spontaneously ignited by the red-hot wheel. This keeps it spinning so that as soon as the throttle is opened, the turbo’s ready to give full boost. And, because the throttle is closed during the above process, the inlet tract is filled with compressed clean air, so that when the throttle is opened, the cylinders are immediately filled and, when fuel is injected, response is instant. The downside to the ALS process is the massive thermal loads it puts on the turbo and exhaust system, and the heat generated within the engine.
Another method to reduce lag (partly) when changing gear is a dump valve. Normally on a gearshift the throttle is closed and the air from the spinning turbo backs up against the closed throttle and stalls it. But a dump valve vents the pressurised air off.