[рони]

noid,
пробуйте, заменить на

if(localStorage.paused !== 'true') playlist();
    else {
        videos[currPlayVideo].classList.add('play');
        videos[currPlayVideo].onended = function () {
            playNextVideo();
        };
        $('.video-control').remove('video-control_active');
    };
    $('.video-control').on('click', function () {
        var currentVideo = document.querySelector('.play');
        $(this).toggleClass('video-control_active');
        if (currentVideo.paused) {
            currentVideo.play();
            localStorage.paused = 'false'
        } else {
            currentVideo.pause();
            localStorage.paused = 'true'
        }
    })