vs
QUICK ANSWER
"Matrimonio" is a form of "matrimonio", a noun which is often translated as "marriage". "Alianza" is a noun which is often translated as "alliance". Learn more about the difference between "matrimonio" and "alianza" below.
el matrimonio(
mah
-
tree
-
moh
-
nyoh
)A masculine noun is used with masculine articles and adjectives (e.g., el hombre guapo, el sol amarillo).
2. (ceremony) (South America)
Regionalism used in South America: Argentina, Bolivia, Chile, Colombia, Ecuador, Paraguay, Peru, Uruguay, and Venezuela
a. wedding 
Me invitaron a un matrimonio este fin de semana, pero no tengo nada que ponerme.I was invited to a wedding this weekend, but I have nothing to wear.
a. married couple 
Vi a un matrimonio viejo besándose en el autobús como adolescentes.I saw an old married couple kissing like teenagers on the bus.
la alianza(
ah
-
lyahn
-
sah
)A feminine noun is almost always used with feminine articles and adjectives (e.g., la mujer bonita, la luna llena).
1. (coalition)
a. alliance 
El rey planea formar una alianza con Francia para atacar a España.The king intends to form an alliance with France to attack Spain.
2. (jewelry)
a. wedding ring 
Escuché que es casado, pero nunca lo vi con la alianza puesta.I've heard he's married, but I've never seen him wear a wedding ring.