Why do a few local areas in Spain pronounce the "z" like "th", while others pronounce it like "s" (just like in the word "influenza")?
It's a phenomenon known as seseo
According to the D.R.A.E:
The second definition match this part of your question: while others pronounce it like "s" (just like in the word "influenza")
As you can see in the Wikipedia (Sorry, it seems that there's no English translation)
La aparición de estos fenómenos data de los siglos XVI y XVII, cuando ocurrió un reajuste consonántico en el idioma.
Which more or less say that such phenomenon originated in the XVI and XVII centuries due to a consonantic readjustment in the language.
In that time, some sounds inherited from Latin dissapeared and part of the Iberian Peninsula "choose" to pronounce z as /th/ and others as /s/ (mainly the south territory). When America was "discovered", ships sailed mainly from such part so the phenomenon spread there.
There is a "complementary" phenomenon: ceceo, replacing the /s/ sound with /th/. Such phenomenon occurs in the same part of the Iberian Peninsula, where fortuitously I'm from. For example, here in Málaga you can hear salchicha (sausage) pronounced "zarcisha", mixing both phenomenons.
I've been reading recently about "seseo" and "ceceo" in the book Una lengua muy muy larga: Más de cien historias curiosas sobre el español by Lola Pons. An interesting book. :-)
The letter "z" is supposed to be pronounced as if "th" in english.
Lets say as if "s" but getting to touch your teeths with your tongue.
Some and/or sometimes Spaniards, may pronounce "z", "s", and even "ci" making them sound kind of "sh"
Why?.. Well i'm native spanish speaker from south America, and i asked a Spaniard, in a spanish chat, i asked: "what's the thing with the "sh"... I thought, well whatever... And the guy's answer was: "you would not understand" LOL
No need to explain more, the guy was just taught to speak that way since a child... For having said that, perhaps it is kind of a fashion thing. Lets wait a spaniard to answer :)