QUICK ANSWER
"In spite of" is a preposition which is often translated as "a pesar de", and "despite that" is a phrase which is often translated as "a pesar de eso". Learn more about the difference between "in spite of" and "despite that" below.
in spite of(
ihn
spayd
uhv
)A preposition is a word that indicates the relationship between a noun and another word (e.g., He ran through the door.).
1. (despite)
a. a pesar de
We went to the baseball game in spite of the rain.Fuimos al partido de béisbol a pesar de la lluvia.
despite that(
dih
-
spayt
thaht
)A phrase is a group of words commonly used together (e.g., once upon a time).
1. (general)
a. a pesar de eso
Our budget has been significantly cut this year, but despite that, we will go ahead with our project.Este año, han reducido nuestro presupuesto de manera significativa, pero a pesar de eso, seguiremos adelante con nuestro projecto.
b. pese a eso
Unemployment is on the rise. Despite that, some positive signals seem to be emerging.El desempleo está en aumento. Pese a eso, parecerían estar surgiendo algunas señales positivas.