Congenital Syphilis: This is the most likely diagnosis given the mother's positive syphilis status. Congenital syphilis occurs when the spirochete Treponema pallidum is transmitted from the mother to the fetus during pregnancy. It can cause a range of symptoms, from mild to severe, including skin rashes, snuffles (a chronic, possibly bloody nasal discharge), hepatosplenomegaly, and neurological problems.

Neonatal Herpes Simplex Virus (HSV) Infection: Although not directly related to syphilis, neonatal HSV can present with similar symptoms such as skin lesions and can be a consideration in infants born to mothers with a history of sexually transmitted infections.
Toxoplasmosis: Another congenital infection that can result from maternal infection during pregnancy, causing a range of symptoms including neurological issues and vision problems.

HIV Infection: Given the mother's history of syphilis, there is an increased risk of HIV infection, which can be transmitted from mother to child during pregnancy, childbirth, or breastfeeding. Early diagnosis and treatment of HIV are crucial for the child's health and survival.
Congenital Rubella: Although less common due to vaccination, congenital rubella can cause severe birth defects and should be considered in the differential diagnosis, especially if the mother's vaccination status is unknown or if she was exposed during pregnancy.

Congenital Lyme Disease: While rare, if the mother has a history of Lyme disease, especially during pregnancy, there is a theoretical risk of transmission to the fetus, although this is extremely uncommon.
Other Congenital Infections (e.g., CMV, Parvovirus B19): These infections can cause a variety of symptoms in newborns, including anemia, thrombocytopenia, and hepatosplenomegaly, and should be considered in the differential diagnosis based on clinical presentation and maternal history.