"Today was raining" x "It was raining today"

Larissa, não há diferença. Entretanto, a primeira sentença precisa de "it" antes de "was".

"Today it rained..."
"Today it was raining..."

"It rained today."
"It was raining today."

Você sempre precisa usar "it" para falar de fenômenos da natureza, de condições climáticas ou da temperatura.

Exemplos:
"It was raining yesterday."
"It's snowing today."
"It's hot in here."

Actually, guys, I don't think you really need a pronoun in cases like

Yesterday was sunny/warm/frosty/freezing/etc.
2012 was warm, but 2011 was very cold.
Today is mild compared to yesterday.
That day was terrible.

But I do agree with "Today it rained/was raining." "Today rained" simply doesn't work. It's so short that you need that pronoun to let the message flow along. (No pun intended.)

You can look at it this way:

Yesterday it rained => Here, "yesterday" is an adverb used to place the said event in time. Since "yesterday" cannot rain, you need the "it" before "rained". You are not qualifying "yesterday" but rather telling the listener what happened yesterday.

Yesterday was cold => Here, "yesterday" is a noun, which you are describing with an adjective. This is much like saying "the house was cold".

In this case, knowing the adverb from the noun can give you a better idea when it is appropriate to use "it" and when it is not.

Hope that helps.

Nossa, Henry Cunha, você está certo. Eu realmente não deveria ter generalizado.
Eu senti a falta do "it" em "Today rained" e disse logo que o "it" sempre seria necessário nesses casos. Nesse caso específico a falta dele faz a frase soar estranha, mas como você esclareceu ele nem sempre é necessário.