Hypogonadism: Given the patient's symptoms of fatigue and erectile dysfunction, along with low free testosterone (250 pmol/L) and total testosterone (16.6 nmol/L) levels, hypogonadism is the most likely diagnosis. The elevated luteinizing hormone (LH) level (10.7 IU/L) suggests primary hypogonadism, where the testes do not produce enough testosterone, leading to an increase in LH secretion by the pituitary gland in an attempt to stimulate testosterone production.

Late-Onset Hypogonadism (LOH): Also known as andropause or male menopause, LOH is a condition where men experience a decline in testosterone levels with age, leading to symptoms such as fatigue, erectile dysfunction, and decreased libido. The patient's age and symptoms align with LOH, although the diagnosis would require further evaluation to distinguish it from other forms of hypogonadism.
Pituitary or Hypothalamic Dysfunction: Although the elevated LH level suggests primary hypogonadism, pituitary or hypothalamic dysfunction could also lead to hypogonadism if there is a problem with gonadotropin-releasing hormone (GnRH) or LH secretion. Further testing, such as assessing other pituitary hormone levels, might be necessary to rule out this possibility.

Pituitary Tumor: A pituitary tumor could lead to hypogonadism by disrupting normal pituitary function, including the secretion of LH and follicle-stimulating hormone (FSH). Although less likely, given the elevated LH level, a pituitary tumor is a critical diagnosis not to miss due to its potential for significant morbidity and the need for specific treatment.
Testicular Cancer: Testicular cancer can lead to hypogonadism, especially if it affects both testes. Given the patient's symptoms and hormonal profile, testicular cancer, although rare, is a diagnosis that should not be overlooked due to its serious implications and the importance of early detection.

Klinefelter Syndrome: This genetic condition (47,XXY) is a rare cause of primary hypogonadism, characterized by small testes, infertility, and often, elevated LH and FSH levels. Although the patient's age makes this a less likely diagnosis (Klinefelter syndrome is typically diagnosed earlier in life), it remains a possibility if the patient has not been previously evaluated for genetic causes of hypogonadism.
Hemochromatosis: This genetic disorder leads to iron overload and can cause hypogonadism by damaging the pituitary gland or testes. Given the patient's symptoms and the potential for hemochromatosis to cause hypogonadism, albeit rarely, it should be considered in the differential diagnosis, especially if other signs of iron overload are present.