In short:
WP have various default fields:
first_name
last_name
display_name
username
On WP you can change display name in order to have username, last_name + first_name, first_name + last_name and more combinations.
BuddyPress has the “name” field that by default is the display_name.
When you change the BuddyPress name, the WP display name will not change, neither the BP displayed name (on profile pages, on the members directory etc. ) or any WP related data will change.
BuddyPress developed an option ( Settings > BuddyPress > Options > “Allow BuddyPress to Sync with the WordPress Profile” ) but it doesn’t work at all, because then, when you change the BP name field, this will affect on three WP fields: display_name, it will erase the last_name field and will change nickname and the name to the same value, the name you inserted. So this will change the BuddyPress displayed name, but you will loose every data about WP users. This is not a bug, looks like this is what the BP developers wanted to do.
With BuddyPress to WordPress full sync you can sync WP fields and BP fields. So if you change the WP first_name it will sync eg. with the BP field name, or if you change the WP last_name field, it can be synced with a BP surname profile fields and vice versa. The display name will remain the one you set on WP since that this is the BuddyPress default behavior.
Then you can implement BuddyPress with a custom function that can display, in the members directory, the first name + last name fields, or the nickname, but this is not ATM what this Plugin is developed for.