Coming from the same kanji as 難（むずか）しい ‘difficult’, 難（がた）い is a 接尾語（せつびご） (suffix) that also means ‘difficult’. However, in the case of がたい, it will be attached to the 連用形（れんようけい） (conjunctive form) of verbs, and conveys that ‘(A) is difficult to do’. がたい is almost always written without kanji in this meaning.

• あいつが言（い）う事（こと）は信（しん）じがたいが、今（いま）は信（しん）じるしかない。
  It is hard to believe what he says, but we have no choice but to trust him.
• 言（い）いがたかったから言（い）わなかったけど、シャツ反対（はんたい）だよ。
  I didn't say this because it’s hard to say, but your shirt is inside out.

Compared to にくい, which can simply imply that something is ‘uncomfortable’ to do, がたい refers almost exclusively to 困難（こんなん）であること ‘things that involve a high degree of difficulty, distress, or infeasibility’. Due to this, がたい will come across as far stronger than にくい, and may imply that a certain task is almost impossible.

• 彼（かれ）の声（こえ）は聞（き）こえにくい。
  His voice is a bit tricky to hear. (It is kind of hard to hear his voice, based on my ability/feeling)
• この話（はなし）が信（しん）じがたいのは分（わ）かるが、信（しん）じてくれ！
  I understand that this story is impossible to believe, but please believe me! (Believing something like this is not easy)

Summed up, these nuances basically translate as the following:
にくい - Something is difficult based on someone’s opinion, experience, or general feeling.
がたい - Something is difficult based on the task itself having a high level of difficulty.