Adding a span when custom post type is updated

No, you’re right. This is the code that is responsible for ‘New’ span:

<?php if (strtotime($post->post_date) > strtotime('-14 days')): ?>
    <div class="new-job-tag"><span>New</span></div>
<?php endif; ?>

The only problem with your modification is that there is no field called the_modified_date in WP_Post object.

The proper name of that field is post_modified, so your code should look like so:

<?php if (strtotime($post->post_modified) > strtotime('-14 days')): ?>
    <div class="updated-job-tag"><span>Updated</span></div>
<?php endif; ?>

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