The dictionary gives both words to mean 'ominous' and they have the same pronunciation. What is the difference between 不気味 and 無気味?
In your question you write "both words", but in fact they're the same word – just written two different ways. Both are correct and have the same meaning.
不気味 is the most common way to write the word today. If I search the Balanced Corpus of Contemporary Written Japanese (BCCWJ) with the 少納言 tool, I find the following results:
However, 124 of those 131 results are in the 書籍 subcorpus, and of those 91 are in the 文学 genre. If I instead check the boxes to only use the Yahoo!知恵袋 and Yahoo!ブログ subcorpora, I find a much starker contrast:
In other words, in casual writing people today tend to just write the word as 不気味, but you can still find 無気味 in published writing, mainly in literature.
However, only 45 years ago the situation was different. From 1948 until 1973, it was taught that this word should always be written 無気味. At the time, the Tōyō Kanji (predecessor to the Jōyō Kanji) only allowed ふ as a reading for 不, which meant that the negative prefix ぶ had to be written 無 or ぶ. However, this reform never fully caught on, and in 1973 the government reversed its decision and allowed ぶ as a reading for 不 as well, restoring 不気味 as a standard way of writing the word.
You can still find 無気味 today, but mostly in literature as an aesthetic choice by the writer. You'll also encounter it if you read pre-war Japanese, for example 思ひ出す事など by 夏目漱石, although 不気味 was used commonly in pre-war writing as well.
Since 不気味 is most common in most contexts, I suggest you write it that way yourself unless you feel you have a reason to do otherwise.