The distinction is not really specific to Barrister and Mann per se, other than Will happens to do a soap of that style I really enjoy. There are a number of Italian style herbaceous/citrus soaps and splashes on the market. Making time to acquire some B&M is something I would very highly recommend...
In posting I thought it was a good opportunity to draw attention to the two broadly speaking, general 'iterations' of barbershop fragrances for the newer members here and how there has been a wealth of discussion regarding them over the years, in short some people associate barbershop with the musk/powdery group while for others it's the Italian style. The general consensus is that 'barbershop' does not have a fixed distinction, it's a catch all term for two different families of fragrances.
If you do some searching on P&C, reddit and across all the other wet-shaving forums for a number of years you will find the same discussion with two main flavours of opinion. Sharpologist (I think) even ran a straw poll and collated a bunch of data based on preferences for the two types in an effort to find out what factors influence preference and response. The TLDR of the data and most of the discussions is that depending on your nationality, age and so forth people will tend to associate the fragrance of 'barbershop' with generally either the musk/powder type (American barbershop) or the Italian type of barbershop. There is no right and wrong, it's down to personal likes and preferences.