Difference: 小气 vs. 小器?
Today they are interchangeable as long as they mean "miserly". 小器 predates 小氣

In The Analects, Chapter 3 (論語 ‧ 八佾) (4th-3rd century BC)


  子曰：管仲之器小哉。或曰：管仲儉乎


"Confucius says Guan Zhong is miserly, but someone asks if he is frugal?" 

Does it make sense that 小器 means miserly? Yes and no because "miserly" and "frugal" are not identical. But 小器 also means narrow-minded or intolerant as if he has small "capacity to contain" (器) different opinions.

In Dream of the Red Chamber (18th century), Cao Xueqin used 小器 and 小氣, both meaning miserly.

In Chapter 40, 


  不送些玩器來與你妹妹，這樣小器。


"You didn't give those toys to your sister. So miserly"

But in Chapter 67


  我又不是兩三歲的孩子，你也忒把人看得小氣了


"I'm not a child of 2-3 years old. You are mistaken that I'm miserly"

Does it make sense that 小氣 means miserly? Yes and no because 小氣 also means petty, narrow-minded and frivolous.

In 漢語大詞典, 小器 and 小氣 are synonyms. In 辭源 (1908), which focus on etymology and classical Chinese, only 小器 is listed as entry with the definition


  小器——吝嗇。同‘小氣’


小器, miserly, same as 小氣

In many modern dictionaries, 小器 is omitted.

In summary, today you can safely use either 小氣 or 小器 (miserly), either 氣度 or 器度 (tolerance). Only 小氣 (simplified: 气) is more modern. Thus, in modern colloquial 小氣鬼 (a scrooge), 氣 is used.