Estrábico vs. vesgo
Both estrábico and vesgo describe the condition where a person's eyes do not look in the same direction simultaneously. The primary difference lies in the register: one is a medical term, while the other is part of everyday vocabulary.
Estrábico
C1This is the formal, clinical term derived from estrabismo (strabismus). It is the polite and objective way to describe the condition in medical contexts, formal writing, or when you want to ensure you are not being offensive.
O médico diagnosticou o paciente como estrábico ainda na infância.
(The doctor diagnosed the patient as having strabismus whilst still in childhood.)
Existem óculos especiais para corrigir o foco de um olho estrábico.
(There are special glasses to correct the focus of a squinting eye.)
O pintor retratou o rei como sendo levemente estrábico.
(The painter portrayed the king as being slightly cross-eyed.)
Vesgo
B1This is the colloquial, popular term for cross-eyed. It is used in daily conversation and informal settings. While it is the most common word, it can sound pejorative or mocking depending on the tone of voice, so it should be used with caution when describing people.
Fiquei vesgo de tanto estudar para a prova de matemática.
(I went cross-eyed from studying so much for the math test.)
Aquele cachorro vesgo é muito engraçado e simpático.
(That cross-eyed dog is very funny and nice.)
Ele fez uma cara de vesgo para fazer as crianças rirem.
(He made a cross-eyed face to make the children laugh.)
Summary
To summarize, use estrábico in medical offices, professional situations, or to be polite. Use vesgo in casual conversations, jokes, or when describing pets, but be careful as vesgo can effectively be used as an insult, whereas estrábico is neutral.