A garage conversion reuses existing shell; an extension adds footprint. They solve different problems, and the right answer usually depends on what you lose as much as what you gain.
Speed and disruption
A conversion is typically a matter of weeks and largely contained within the garage. An extension involves groundworks, weather risk and a longer programme.
What you give up
Converting removes storage and, on some properties, covered parking. On estates where garages are the only workshop and bike store, that trade-off matters.
Planning and regulations
Conversions are usually permitted development with a building notice. Extensions are more likely to need an application and always involve building control.
Doing both
Converting the garage and extending to the rear is a common combined project, giving a much better ground-floor layout than either on its own.