In ancient times, 这 is often used in colloquial language, while 此 is often used in black and white.
Nowadays 这 is much more popular than 此 since the written Chinese is much more colloquial than it is in ancient times. But some usages of 此 still obtain.
Generally speaking, 此 is more formal and briefer. Based on this principle, we can find some fine distinctions as follows:
1 for some casual words used in spoken languages, we use 这. And 此 is not ok here.
e.g. 这里,这些,这会儿,这个,这么,这样
2 when they both mean 'this', when you use 此 instead of 这, you often leave out certain characters. And in this case, both 这 and 此 are at the begining of the word.
e.g. 这本书,此书;这件事情,此事;这条路走不通,此路不通;这个地方,此地……
3 when they mean more than "this", for example, this moment, right now, they are not exchangeable.
e.g. 我这就去。我这才看见。
从此。就此结束。
此 can mean "this place" too.
e.g. 就此告别。 由此往西。到此一游。
此 can mean more than that... and it usually used after certain characters to form a word..
为此(为了这个目的,因为这个原因),因此(因为这个原因),自此之后(自从这件事以后),照此(按照这个办法,根据这个指示)……