I recently came across a sentence mentioning 'ジャパニーズデザイン'. What would be the difference between calling something 日本のデザイン verses ジャパニーズデザイン?
There are various reasons to use katakana or loanwords (see other answers), but in this case, I think the reason is very close to why 日本 is sometimes written as ニッポン or why トヨタ is written in katakana. The intentional use of the katakana version of a word can make it sound modern and/or internationalized. For example, 忍者 is a plain word for ninja, but ニンジャ written in katakana suddenly makes people imagine that stereotypical ninja enjoyed by western people. Likewise, while 日本のデザイン is the normal way to say "Japanese design", ジャパニーズ・デザイン can be intentionally chosen to give the impression of "that stereotypical Japanese design appreciated internationally" or "contemporary Japanese design as opposed to universal or traditional ones".
There are a couple of situations where using カタカナ語 (Western loanwords transcribed in katakana) is preferred.
It is cool.
Young people like to say things differently.
The word/phrase was borrowed straight from English.
I think your word falls under this category. It came directly from English phrases such as "American design", "Japanese design."
It's fun to toy with English words and phrases and construct new カタカナ語 (和製英語) in accordance with Japanese grammar simply because they make sense to Japanese speakers with knowledge of English. One example that I recently encountered is ロケハン (location scouting). And choosing them over other older Japanese words goes back to situation #1: it's cooler to use those English-sounding words.
Some カタカナ語 have come to carry nuances that set them apart from traditional words (kango and wago). For example, チャイニーズ (Chinese) has increasingly been used to refer to Chinese cuisine and restaurants (中国料理), as opposed to 中華料理.
At the first sight of this expression ジャパニーズデザイン, I felt it very strange as Japanese English, because of its relative length. This leads me into some search of this expression on the Web. I have found some facts about this expression. Firstly, as a native speaker of Japanese, anyone would not like to use 日本のデザイン, except in translation. We will probably prefer to use 日本風のデザイン or 日本式のデザイン or 日本特有のデザイン or 日本独自のデザイン or 日本人によるデザイン, even though these are all explanatory. Secondly, through the examination of about ten uses of ジャパニーズデザイン found on the web, I have discovered that this is almost used in connection with the world, especially the western world and is referred to the things of Japanese designed with western taste, or the western things with Japanese taste.