So I am assuming that you are asking primarily why visigoths came to abandon that identity in favor of another rather than asking why different regions of Spain have different identities.

The visigoths came to Spain in a way somewhat different from other barbarian groups coming into the late roman empire. The visigoth had already been wandering throughout the empire, sometimes rebelling, for many years before they arrived in spain and the emperor Honorio ceded the country to them. (Some see this as a sign of weakness but it may have prevented a lot of bloodshed and ultimately helped spain avoid the same degree of destruction in the rest of the empire.)

Since they had been in the empire for so long, the visigoths were already very influenced by roman culture. While the lombard and franks both left their languages in certain regions they conquered (belgium, isolated german dialects in northern italy), the visigoth language has left no descendants today and even seems to immediately disappear upon the visigothic arrival in spain. Anothe piece of evidence is the relative lack of visigothic words on the Spanish language. They are so few in fact, as to be negligible in the study of the language. Only certain vocabulary words attest the existence of this people in spain.

Finally, the visigothic identity's dis appearance would have been more or less complete with the promulgation if the Libro de los Juicios, a law code enacted by King Chindasuith. With it, the inhabitants of the Visigothic kingdom were all to be called Hispani, and not goth or roman.