At the simplest of levels both mean relatively the same thing. However, the nuance - and thus times you may choose one over the other - is slightly different. 離す means more to separate from something, where 放す means to release something (the clearest image is "out into the wild" or "on its own").
For reference, from the Microsoft IME:
- 離す → 分離，距離 「二人の仲を離す，つないでいた手を離す」
- 放す → 開放，自由 「鳥を放す，ハンドルから手を放す」
As a side note, as a literary technique you can also purposely mis(?)-use kanji in these cases to convey a slightly different meaning or nuance in written text. That is, one might write 彼女の手を放した to convey the feeling that he permanently let her go, with an added feeling that the woman as an entire person left, the hand simply being the device of expression.
Furthermore, songs and poems may even use non-standard or unusual kanji entirely to convey ideas, generally accompanied by furigana - one simplest example being
時間【とき】. In this example (which would normally be read
じかん), the idea of "a long time" or "time passing" is clearly conveyed, as opposed to a single moment in time.