Desabotonar vs. desabrochar
The Spanish verbs desabotonar and desabrochar both relate to undoing clothing and accessories, but they differ significantly in their scope. While desabotonar is highly specific to a single type of fastener, desabrochar serves as a general umbrella term for unfastening almost anything.
Desabotonar
The verb desabotonar translates specifically to to unbutton. It is derived from the noun botón, which means button in English. Consequently, this verb is strictly used when you are undoing the buttons of a garment, such as a shirt, a winter coat, or a cardigan. It cannot be used for any other type of fastening mechanism.
Hacía mucho calor, así que decidió desabotonar su camisa.
(It was very hot, so he decided to unbutton his shirt.)
Tienes que desabotonar el abrigo antes de quitártelo para no romperlo.
(You have to unbutton the coat before taking it off so you do not tear it.)
Me cuesta desabotonar los pantalones cuando tengo las manos frías.
(I find it hard to unbutton my pants when I have cold hands.)
El niño pequeño por fin aprendió a desabotonar su chaqueta sin ayuda.
(The little boy finally learned to unbutton his jacket without help.)
El sastre empezó a desabotonar el chaleco del cliente para tomarle las medidas.
(The tailor started to unbutton the client's vest to take his measurements.)
Desabrochar
The verb desabrochar translates to to unfasten, to undo, or to unclasp. It originates from the noun broche, meaning clasp or fastener. This is a much broader and more versatile term. While it can be used to mean unbuttoning a shirt, it is mostly used as the correct word for undoing seatbelts, zippers, buckles, snaps, hooks, bras, and necklaces.
No debes desabrochar el cinturón de seguridad hasta que el avión se detenga.
(You must not unfasten your seatbelt until the plane comes to a stop.)
Ella se apresuró a desabrochar la correa del perro cuando llegaron al parque.
(She hurried to unfasten the dog's leash when they arrived at the park.)
Es mucho más fácil respirar después de desabrochar el cinturón del pantalón.
(It is much easier to breathe after unfastening the belt on the pants.)
Se me rompió la uña al intentar desabrochar el collar de oro.
(I broke my nail trying to unclasp the gold necklace.)
El doctor le pidió al paciente desabrochar su bata para poder examinarlo.
(The doctor asked the patient to undo his gown in order to examine him.)
Ayúdame a desabrochar la mochila porque el cierre de plástico está atascado.
(Help me unfasten the backpack because the plastic buckle is stuck.)
Summary
In summary, desabotonar is strictly limited to undoing physical buttons, whereas desabrochar is a versatile verb used for unfastening any kind of closing mechanism, including seatbelts, zippers, clasps, and buckles. You can often use desabrochar in place of desabotonar when talking about clothes, but you can never use desabotonar to talk about unfastening a seatbelt or a dog leash.