Q:

Why is it "el arte moderno" and "las artes finas"? 

(I'm a native Spanish speaker.) 
Why the apparent change of gender? 

A:

arte can be either feminine or masculine in any sense, per the DPD:

Este sustantivo, que era femenino en latín, puede usarse en ambos géneros, teniendo en cuenta que, en el español actual, cuando va en singular, lo normal es usarlo en masculino […] y, cuando va en plural, lo normal es usarlo en femenino. 

The reason no doubt is related to the fact that arte takes el as its singular article no matter what, so after enough uses of el arte(fem.)  contrasted with las artes(fem.),  some people started reinterpreting the singular as el arte(masc.).    Words that  began with /a/ underwent some interesting changes especially during the period in which the rules for when you use feminine article el were in flux. 
After a period of confusion, arte settled with the sort of uncountable-ish use in masculine, and when used in a more skill-based sense, being feminine.   Because the former is used normally in singular and the latter more often in feminine, the association of masculine-in-singular and feminine-in-plural came about, but it's not strict or universal: el arte posmoderno es interesante mientras los artes vanguardista y moderno me aburren; poder escribir poesía es un(a) arte literaria.   So basically, the confusion continues today, permitting use in either gender.