what makes them different is limited by the game mechanics.
the BHG can follow a suspect into the badlands and up to rochester and be in RP. - the LPI can NOT... cause regarding to the lore, they do not dare. ( they are given a paycheck by the end of the month, no matter how many they catch )
the BHG takes greater risks - the LPI does what it "must" do. what sets the LPI apart from lets say ... the kusari state police? - the LPI has nothing you d call "patriotism" or "honour". and right so... they are employed to do a job, they do the job - and nothing more.
a KSP would go MUCH further than "just doing the job" - a BHG would go further cause he MUST do this deed. - a rheinland police officer would take greater risk, cause they do it for their mothercountry and their chancellor... a greater good, if you want.
by putting the LPI to be a private organization. - it makes them somewhat down to the common employee. - they are no heroes, cause they are not thanked for being one. - they are no real patriots, cause they are not liberty.
but putting the LPI to be private also removed them from the goverments responsibility. - if the LPI is corrupt, its only the LPI - not the liberty gov. - if they are lazy and weak, its not the liberty goverment, but only the LPI.
if the rheinland police was weak and cowardly, it would reflect upon rheinland - a great disgrace. - if a BHG is a coward, - he ll risk chances to get contracts. - the only policeforce in all sirius that can actually do the minimum of whats expected from them is the LPI.
how is that reflected in the game? - it is not... simple as that. - but its not the only faction.... most factions are improperly reflected. - many a factions have deep lore and great RP opportunities, but when you get down to it.... all you see from them ingame is some rather low key pvp.
the game simply does not allow all the fascettes of a faction, not even such a simple one as the LPI.