The difference is that this is used to refer to things that are nearby, either in time or space or even emotionally, while that is used to refer to things that are more distant. Obviously, it is not always clear whether something is "nearby" or "distant", especially if it's not something physical, and this is why you'll sometimes see the words used interchangeably.
Let's consider your example sentences.
The weather in Seattle is different from that in New York.
In this case, I would use that, because "the weather in New York" is a general concept that does not exist right here and now; we could say this whether we were in Seattle or New York or Shanghai or anywhere. "This" would imply "this weather that we are experiencing right here and now in New York".
I feel ill. I've got a stomach ache. -- How long have you been like this?
In this case, I would use this, because the illness and stomachache are something that the speaker is experiencing right now.
To give some other examples, if it were snowing right now in New York, I might gesture to it and say,
The snow in Denver is much heavier than this.
because the snow I am referring to is right here and now. Or I might say
It rarely snows in New Orleans. This is because it's much farther south.
In this case I'm using this because I just referred to the fact, so it is current, or close in time.
If someone said to me
Last year I was sick to my stomach for a long time
I might ask
How long did you feel that way?
because the illness is distant in time.