У тебя asNavFor: '.slides-dots', а класс слайдера .slider-dots.

$(function(){

$('.header__slider').slick({
  infinite: true,
  fade: true,
  prevArrow: '<img class="slider-arrows slider-arrows-left" src="img/arrows-left.svg" alt="">',
  nextArrow: '<img class="slider-arrows slider-arrows-right" src="img/arrows-right.svg" alt="">',
  asNavFor: '.slider-dots',
});

$('.slider-dots').slick({
  slidesToShow: 4,
  slidesToScroll: 4,
  asNavFor: '.header__slider',
});

});