Sort of related. Hashimoto's is an autoimmune thyroid disease (the body attacks itself) which can cause hypothyroidism. Hypothyroidism, however, can be caused by other etiologies and not just hashimoto's.
One leads to another. The former can lead to the latter. The opposite is not the case however. Additionally, hypothyroidism is a clinical state whereas hashimoto's is in essence a histological, descriptive term based on the findings of immune cell infiltration of the thyroid gland tissue first described by dr.Hashimoto, a japanese family physician.