"why do I need a flue pipe adapter ?"
Antique French & Belgian stove flue outlets/spigots came in all sorts of wonderfull shapes and sizes, and to this day 3 or 4 sizes are still common on modern stoves!
Flue pipe in the older diameters is becoming very difficult to locate in enamelled or stainless steel, and these are the only two materials allowed to connect to a solid fuel stove in the UK. So modern pipe diameters are now used for antique stoves in sizes that are easily available from all stove pipe stockists. 100mm (4"), 125mm (5"), 150mm (6") etc etc.
Therefore if you have a 1920's Godin stove with a flue outlet of say 82 - 84mm taper, you may want to connect into a modern 100mm flue pipe, and the only way to do this correctly is to use an adapter similar to the examples at the top of the page.
On occassion some stoves will have a flue outlet very close to modern sizes, for example a tapered outlet with the dimensions of say 94 - 99mm. In this case a modern flue pipe with an internal diameter of 97-99mm will be a snug fit over the stove outlet and no adapter should be required, but... do check the internal diameter of the pipe bend or Tee before purchase as usually the dimensions given will be for the pipe outside diameter leaving the internal diameter around 2 - 4 mm smaller.