Muscle Strain or Contusion: The patient's history of falling on the elbow while playing with his kids, followed by pain in the left bicep that radiates to the forearm and is exacerbated by activity, suggests a muscle strain or contusion. The mechanism of injury and the nature of the pain are consistent with this diagnosis.

Medial Epicondylitis (Golfer's Elbow): Given the patient's symptoms of pain radiating to the forearm, which could involve the medial aspect of the elbow, and the onset after a fall, medial epicondylitis is a plausible diagnosis. However, the primary complaint of bicep pain might make this less likely than a direct injury to the bicep or surrounding structures.
Bicipital Tendinitis: Inflammation of the biceps tendon could cause pain in the bicep area, especially if the patient's fall caused direct trauma or strain to the tendon. The pain pattern and exacerbation with movement support this possibility.
Radial Neuropathy or Radial Nerve Compression: The radiation of pain to the forearm could suggest involvement of the radial nerve, especially if the patient's fall caused compression or injury to the nerve.

Fracture (e.g., Radial Head or Neck Fracture): Although the patient does not report significant swelling or deformity, a fracture could still be present, especially given the mechanism of injury (fall onto the elbow). Missing a fracture could lead to improper healing and long-term disability.
Compartment Syndrome: This is a medical emergency that could result from increased pressure within the compartments of the forearm, potentially caused by trauma from the fall. While less likely given the time frame and description, it's crucial not to miss this diagnosis due to its severe consequences.
Vascular Injury: Injury to the brachial artery or its branches could occur from a fall onto the elbow, potentially leading to ischemia or other vascular complications. This would be a rare but critical diagnosis not to miss.

Osteochondritis Dissecans of the Capitellum: This condition involves a piece of cartilage and bone becoming detached from the capitellum, often due to repetitive stress but could be precipitated by a traumatic event. It's more common in young athletes but could be considered in the differential for elbow pain after a fall.
Infection (e.g., Osteomyelitis or Septic Arthritis): While unlikely without other symptoms such as fever or significant swelling, infection is always a consideration in the differential diagnosis of acute limb pain, especially if the pain is severe or not improving with rest.