Maximum approval with Danse awards the Know Your Enemy perk which grants +20% more damage against ghouls, super mutants, and synths.
Danse is a potential candidate for romance, an option that unlocks after reaching the highest level of affinity.
It is impossible to max out the relationship with Danse before completing Blind Betrayal.
Siding against the Brotherhood before Blind Betrayal turns him hostile, removing him as a companion.
Siding against the Brotherhood after Blind Betrayal results in Danse no longer interacting with the Survivor. However, the player still gets the Lover's Embrace bonus by sleeping near him, which appears to be a bug. The issue effectively removes Danse as a companion but it will not remove his perk.
To keep Danse as a full companion following the main story one must either A) side with the Brotherhood, or B) side with the Minutemen and choose not to become enemies with the Brotherhood of Steel. The Railroad and Institute endings both force the player character into conflict with the Brotherhood.