What is the difference in usage between 「やってみます」and 「試します」？
For example, what is the difference when you say 「やってみてください」and「ためしてください」
to ask someone to try something out

Question

What is the difference in usage between 「やってみます」and 「試します」 ？
For example, what is the difference when you say 「やってみてください」and「ためしてください」
to ask someone to try something out

Answer

***** Brief conclusion

"試す／試してみる" and "やってみる" share the same basic meaning, "try".

The diffrence is:
試す： try under control
やってみる: try not under control

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**** する vs やる
する： do (rather carefully, with thinking)
やる：do (roughly / without thinking)

OK：支度をしてください （get prepared)
NG：支度をやってください

「やっちまえ」：beat him out 
「やりっぱなしじゃダメでしょ」： Once you get started, you must finish.

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***  about "やってみる"

= 「やって（=　やり+て）」＋「みる」

やって： do (without thinking), and
みる： see

やってみよう：　
Let's try, or Let's do it without thinking too much and see what happens.

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***  about "試す"

"試す" also means "try", "do it and see  what happens".
And when you "試す" something, you know in advance the standard/criteria to judge the result, whether it is good or bad, 
but when you "やってみる" something, you don't have such standards.

「試す」is like put something "on the TRIal"
「やってみる」is simply "do it!"

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**** The biggest difference in usage.

The  verb "試す" can have the objcets which indicates "quality", while "やってみる" not.

OK：「真価を試す / 真価が試される」
       　(do something, and see it has real value or not)
NG：「真価をやってみる」

OK：「あいつを試す」 
       　(do something, and see he is good or not)
NG: 「あいつをやってみる」

-「切れ味を試す」: (cut something using the sword/knife/scissors, to find how well it can cut)

- 「服を試す」：wear clothes before you buy to see if it goes good with you or not.

-「商品を試す／商品を試しに使う」：
(use the product, to find how good it is for you).

-「能力を試す」
(let him/her do something, and see how he/she is able).

**** FYI:
「試し」originally meant in old Japanese "a (good) example".

**** FYI2:
you can say 「試しにやってみる」, do it for a try.

This implies "endless thinking gets you nowhere, now just do it , then you'll see it is good or not"

Answer

やってみる　is informal compared to  試してみる.
So, we don't use　 やってみる　in document such as computer manual.

Answer

Look alike but different. For example.

OK：さあ、モンゴル語をやってみよう！
NG：さあ、モンゴル語を試してみよう
            （Syntactically OK but sematically this sentence can be said in so a limited situation).

I'll add reasons and explanations later.

Answer

やってみる：try, have a try, take a chance
試す：make a valuation, assess, test

Answer

ブティックなどで店員が

どうぞお好きな服をお試しください

とは言うことはありますが、”やってみる”は使えません。
”やる”は、英語のdo ですが、これは動詞としては限定されたものです。
それ以外の動詞の場合は使えません。万年筆を試し書きしたい場合も同様です。