Hand, Foot, and Mouth Disease (HFMD): This is the most likely diagnosis due to the characteristic involvement of the soles of the feet, hands, and sometimes the armpits with painful vesicular rashes. HFMD is a common viral illness, typically caused by coxsackievirus A or enterovirus 71, and it predominantly affects children.

Herpes Simplex Virus (HSV) Infection: HSV can cause painful vesicular rashes and can involve unusual sites like the hands and feet, although it's less common. The presence of fever, lymphadenopathy, and the appearance of the vesicles could help differentiate it from HFMD.
Varicella-Zoster Virus (VZV) Infection: Although primarily known for causing chickenpox, VZV can reactivate as herpes zoster (shingles) and cause a vesicular rash. However, the rash typically follows a dermatomal distribution, which might not exactly match the described locations.
Eczema Herpeticum: A disseminated herpes simplex infection occurring in patients with atopic dermatitis, characterized by widespread vesicular rash, which could potentially involve the hands, feet, and armpits.

Smallpox: Although eradicated, in the event of a bioterrorism attack, smallpox could present with a vesicular rash that progresses through stages, including involvement of the palms and soles. The severity and potential for outbreak make it a critical diagnosis not to miss.
Secondary Syphilis: Can cause a rash on the palms and soles, which is a distinctive feature. While not typically vesicular, the rash can be maculopapular or papulosquamous, and the disease's potential for serious complications if untreated makes it a "do not miss" diagnosis.

Enterovirus 71 Infection: While similar to HFMD, certain strains like EV-71 can cause more severe neurological and systemic complications. The rash itself might not differ significantly from HFMD, but the potential for severe disease warrants consideration.
Dyshidrotic Eczema (Pompholyx): Characterized by small, itchy vesicles or blisters on the hands or feet, it's more common in adults and can be triggered by various factors including allergies or stress. The lack of systemic symptoms and a more localized distribution might distinguish it from the other diagnoses.
Keratoderma Blennorrhagicum: Associated with reactive arthritis, it presents with vesicular lesions on the palms and soles, which later become hyperkeratotic. This condition is rare and usually accompanied by other symptoms like arthritis, urethritis, or conjunctivitis.