Acercarse vs. aproximarse

The Spanish verbs 'acercarse' and 'aproximarse' both mean 'to approach' or 'to draw near to'. However, their usage can often depend on the context.

Acercarse

'Acercarse' generally means 'to move closer' or 'to get near to'. This word often indicates some sort of physical movement towards something or someone.
Voy a acercarme a la ventana para ver mejor.
(I'm going to move closer to the window to see better.)
Él se acercó a y me dijo al oído que tenía un secreto.
(He drew near to me and whispered that he had a secret.)
¡No te acerques al fuego!
(Do not approach the fire!)

Aproximarse

'Aproximarse' can also mean 'to approach' or 'to draw near'. However, this term is often used in more abstract senses, like time or similarity, rather than physical movement.
Se aproxima el fin de semana.
(The weekend is approaching.)
Ella se estaba aproximando a la verdad.
(She was getting close to the truth.)
El número de visitantes se aproxima a mil.
(The number of visitors approaches one thousand.)

Summary

'Acercarse' and 'aproximarse' both refer to the act of moving closer or drawing near. The difference lies in their context of use. While 'acercarse' typically denotes physical movement towards something or someone, 'aproximarse' is often used in more abstract terms such as time, numbers, and concepts.