目の色が違う猫
Here, Spanish is in an opposite order to Japanese. Where in Spanish you might say "Un gato con ojos de colores diferentes" (A cat with different colored eyes), Japanese normally places its adverbs, adjectives, and noun modifications before the noun. In a literal sense, think of this structure in this form:
目の色が違う猫
"eye's colors are different" cat
"el color de los ojos es diferente" gato
This is somewhat closer to English's construction of placing descriptions before nouns. If we change the descriptive sentence of 目の色が違う to something simple like 黒い, for example, this might make a bit more logical sense:
黒い猫 -> "black" cat (un gato negro)
白い猫 -> "white" cat (un gato blanco)
きれいな猫 -> "clean/pretty" cat (un gato limpio/bonito)
Japanese continues this sort of logical construction where in English it changes when adding verbs, using the "that" or "which" connection:
走っている猫 -> a cat which is running (un gato que corre)
寝ている猫 -> a cat that is sleeping (un gato que duerme)
... or using the "with" connection when talking about features:
尻尾が白い猫 -> a cat with a white tail (un gato con cola blanca)
目が青い猫 -> a cat with blue eyes (un gato con ojos azules)
目の色が違う猫 -> a cat with different colored eyes (un gato con ojos de colores diferentes)
EDIT: As for the other sentence structure you mentioned:
猫の目の色が違う
This is a simple, standard sentence. It means "The eye color of the cat is different/wrong/changed"
Let's parse both sentences to see why this is:
[ (目の色) が違う] 猫
[ (eye color) is different ] cat
[ (color de ojo) es diferente ] gato
This is describing the cat. The cat's eye colors are different (from one another).
[ 猫の (目の色) ] が違う
[ cat's (eye color) ] is different
[ del gato (color de ojo) ] es diferente
This is describing the difference. What is different? The cat's eye colors (not from one another, but from something else or a different time.)
EDIT 2: Added Spanish into the examples.
The cat's eye(s) is(are) different
as in it changed. The second is that the cats eyes are different colors. I think...