Mismo vs. igual
In the Spanish language, 'mismo' and 'igual' play unique roles even though they both translate to 'same' in English. The difference lies in how they're used in sentences and their relationship to the words or concepts they modify.
Mismo
'Mismo' roughly translates to 'self' or 'same' in English. The word is usually used to refer to the same person or thing mentioned earlier.
Estoy viendo el mismo programa de televisión que ayer.
(I'm watching the same TV program that I did yesterday.)
Vivo en la misma casa desde que nací.
(I have lived in the same house since I was born.)
Juan se culpa a sí mismo por lo sucedido.
(Juan blames himself for what happened.)
Igual
'Igual' can mean 'equal', 'identical', or 'the same as'. It's used when two or more things are alike, equivalent, or have equal status or nature.
El vestido es de color azul, igual que tus ojos.
(The dress is blue just like your eyes.)
Todos los estudiantes tienen derechos iguales en la escuela.
(All students have equal rights at school.)
Tiene una actitud igual a la de su madre.
(He has an attitude identical to his mother.)
Summary
While both 'mismo' and 'igual' can be translated as 'same', they have different uses in context. 'Mismo' refers more directly to something previously mentioned, indicating that it's exactly the same thing or person. On the other hand, 'igual' is used to highlight likeness, equality, or similarity among things, people, situations etc.