Meaning of "~~" at the end of a sentence
Non-Standard Usage

Is this an informal exclamation or an emphasis mark?


No, it's neither an exclamation nor an emphasis. It's used for


[Informal] Showing a casual, lively mood. This is the situation of your example sentence. In fact, you can put this symbol freely if you are not so serious, and the number of waves is not strict.


  事实上～只要你不需要很严肃，你可以随意地使用这个符号～～～而且数量也是随意的～～
  
  [In fact ~ you can put this symbol freely if you are not so serious ~~~ and the number of waves is not strict ~~]

[Not in standard, but considered formal] Illustrating the continuation of the voice (suggesting the cadence of the voice). Usually, it is used in the double wave form taking place of two characters, i.e. ～～, and is written vertically middle of a line. For example,


  喂～～ [Hey~~]


Though this usage isn't listed in the official standard, interestingly, in the famous primary school's text, Lu Xun's 《从百草园到三味书屋》, there is a sentence:


  铁如意，指挥倜傥，一座皆惊呢～～；金叵罗，颠倒淋漓噫，千杯未醉嗬～～


And in some other essays and novels


  大男孩的胳膊给老妇人抱住，不能取那翡翠簪儿，“哇～～”突然哭起来了。（叶圣陶《夜》）
  
  “咳～～开～～啦！”桩家揭开盒子盖，也是汗流满面的唱。（鲁迅《阿Q正传》）


You can call it 象声号 (《标点符号使用手册》 has included this usage). So, as many famous writers use ～～ in their works, considering it informal may be controversial. 
[Not in standard, but undoubtedly formal] Standing for the mentioned character in a dictionary (seen in most authoritative dictionaries, like 新华字典). In some reference, it is called 代替号(replacement symbol). This should be a formal usage, but I don't know why the standard hasn't included it.


  省 shěng  ㄕㄥˇ
  
  
  地方行政区域：～份。～会。
  



Standard Usage

The standard usage of this punctuation is a connective, called 浪纹(wave symbol). See the standard 《中华人民共和国国家标准标点符号用法》. 


  4.13 连接号
  
  4.13.1 连接号的形式为“－”，占一个字的位置，连接号还有另外三种形式，即长横“－－”（占两个字的位置）、半字线“-”（占半个字的位置）和 浪纹 “～”（占一个字的位置）。


For example


  亩产1000公斤～1500公斤 [from 1000 kilograms to 1500 kilograms]






EDIT1: definition of "formal" and the very informal usage

Thanks for @fefe's comment. To make it more clear, let me clarify the definition of formal in this answer here.

The word formal in this answer means grammatically formal in punctuation, but NOT suggesting wording or phrasing formal.

In the non-standard usage,


Case 3: It's the de facto standard in Chinese dictionaries. Personally I cannot imagine saying "this punctuation in dictionaries is informal" :)
Case 2: the punctuation "illustrates" the continuation of the voice, so that is to say, you were writing some voice in your works. Such a sentence is usually in a colloquial circumstance, so to this extent, the whole sentence is "informal". However, it's related to the wording and phrasing, but not the punctuation itself. So I said the punctuation is still grammatically formal.


Now let's go further ~~ I think this topic would be related and interesting ~~

On the Internet, people aren't so aware of the grammar. Or, we can even say, the netizens are creating new grammar, though there's a debate resulting in the rule that you should not use the Internet grammar in your Chinese examinations. In fact, there is a much more informal usage of ~, which is so informal that I consider even not to list it in the non-standard usage list. It is widely used in emoticons in China and Japan. For example


  ~~~^_^~~~ [laughing to tears. ~ stands for tears.]
  
  (～ o ～)~zZ [sleeping. ~ stands for closed eyes and the snore.]
  
  ~\ (≧▽≦) /~ [hooray! ~ may stand for something waving in the hands.]






EDIT2: brief history of the establishment of punctuation standard

History of the official standard of modern punctuation in mainland China.


  1919 Some renowned scholars (馬裕藻, 朱希祖, 錢玄同, 劉復, 周作人, and 胡適) submitted a proposal for promulgating the modern punctuation standard (《請頒行新式標點符號議案》).
  
  1920 The ministry of Education of the Northern Warlords government promulgated Order for Adopting Modern Punctuation (《通令采用新式标点符号文》). It is the first standard for modern punctuation published in the form of law.
  
  1951 The General Administration of Press of the PRC government promulgated Usage of Punctuation (《标点符号用法》).
  
  1990 As the punctuation had changed a lot in the past forty years, Press and Publication Administration promulgated the revised version of Usage of Punctuation.
  
  1995 State Bureau of Technical Supervision promulgated the National Standard for the Usage of Punctuation (《中华人民共和国国家标准标点符号用法》 GB/T 15834-1995). On June 1st, 1996, this standard was executed. It is the standard we are using today.


Punctuation and language are changing gradually all the time. I personally believe there will be some day that some of the informal usage of ~ will become formal standard ~~



EDIT3: "Which tilde should I use?"

The wiki page of "tilde" points out there are many kinds of tildes. Because your browser may not correctly render all the characters on that page. I illustrate the table as a picture here (open it in a new page to see full size):


So, according to the standard (providing the size and the position of the 浪纹), in formal text, you should use unicode U+FF5E, i.e., ～. In informal usage, a simple tilde on the keyboard U+007E, i.e. ~, is just OK.