It's raining today or it's rainy today?

The meanings are as expected

• It is raining today can mean that it is raining now and it implies that you expect more of it and it can imply that it had been raining previously in the day - expresses general weather condition (it can be used to express future)

• It is rainy today means that it had been raining or that you expect it to rain; probably it is also raining at the time of writing - expresses general weather condition

• It is raining now means that it is raining right now and might continue or might not

Regarding the diary or journal specifically, you might consider also to express it as past

• It was raining the whole day is pretty clear

I wanted to add one difference between the two phrases that I think has not been covered.

It's raining today.

Raining is a verb, describing the action of rain.

It's rainy today.

Rainy is an adjective, describing what the weather is like today.

Sunny and cloudy are also adjectives that describe the weather, so for parallelism, it makes sense to say "It's rainy today" if you would otherwise write "It's sunny today."

Compare this with "It's raining today" and "The sun is shining today." (Since we can't say "It's sunning today.")