What is commonly referred to in the USA as tuckpointing should be referred to as repointing. That is replacing the old damaged/missing mortar with new mortar. Every effort is made in matching the color and look of the existing mortar. We primarily use what is called "Type O" which is sufficiently soft to be compatible with the historic brick and stone.
Walking around any downtown in the area you can look up and see examples of bad pointing (Dark grey mortar, sloppy finish). In addition to looking bad the hard mortar will harm the surrounding brick leading to issues such as brick spalling and further deterioration of the good mortar behind the bad pointing. Further problems and cost will be incurred when it fails, as removing it is a challenge and the surrounding brick will likely have to replaced to re-laid.