The difference, in English, would be as follows:
'Kore' cannot precede a noun. It's for talking about something without actually mentioning the noun. Example: "This is silly." "This is so much fun." "This is delicious." You would use 'kore' in all of the equivalent Japanese sentences.
'Kono' on the other hand MUST precede a noun. It's when you are referring to an object specifically. Example: "This movie is silly." "This game is so much fun." "This pancake is delicious."
In Japanese:
"This is so much fun."
'Kore ha totemo tanoshii desu.'
You *cannot* say 'Kono ha totemo tanoshii desu.'
"This game is so much fun."
'Kono geemu ha totemo tanoshii desu.'
You *cannot* say 'Kore geemu ha totemo tanoshii desu.'
In short: 'Kore' is "this." 'Kono' is "this noun," where you MUST name the name.
'Kono yume ga owaru to'
kono nihongo ga rikai dekimasuka
You write it as: "この 週末" and it is spelled as "Kono shuumatsu", "kono" meaning "this", "shuu" "week" and "matsu" "end".
You write it as: "この 週末" and it is spelled as "Kono shuumatsu", "kono" meaning "this", "shuu" "week" and "matsu" "end".
この自伝 Kono jiden
kono kyokku ii ne
'Kono eiga ga daisuki desu.'
I believe you are actually writing "kono yakusoku". It is a Japanese phrase which contains a modifier and a noun Kono is the word for "This" referring to the noun it is modifying. Yakusoku means "Promise" or "previous engagement" 約束 【やくそく】 - Yakusoku この - Kono
Dansu kono bīto ni awasete
This-kon example: This coming-kono This evening-kondo no This month-konban This morning-kongetsu
If the picture in question is a photograph, you may say: 'Kono shashin ga daisuki desu.' If it is a drawing/painting/etc., you may say 'Kono e ga daisuki desu.'
Kono hōmupēji e yōkoso (read: kono homu peji eyo koso) このホームページへようこそ