Q&A: Why does Hebrew have multiple spellings for the same word?

Typically, it doesn’t: by and large, there is just one way to spell any given word. There are three types of exception, though:

• ktiv malé (‘Full spelling’) versus ktiv ḥaser (‘deficient spelling’): When writing without the dots and dashes that indicate vowel sounds (which is usually the case, except in texts for little children, poetry, and Scripture), one often uses the letters yod to indicate a /i/ sound (e.g. אימא instead of אמא), or vav to indicate /o/ or /u/ (e.g. נוח instead of נח).
• Foreign words or names are often subject to variations—thus, Turkey can be תורכיה or טורקיה, music can be מוסיקה or מוזיקה.
• Varied spellings of native Hebrew words. These exist, but are very rare, and usually for specific historical reasons—such as תפס versus תפש. These are tolerated because both have precedents in the Hebrew Bible, when spelling was more flexible than today. There are also rare instances where a change of spelling is indicative of a nuanced difference of meaning—e.g. the word etched is normally spelled חרט, but חרת if it’s about etching in memory.

All in all, however, Hebrew spelling is remarkably consistent—far more than English, even without the differences between British and American spelling.

However, it is possible that your question refers to the multiple spellings of Hebrew words in English and other Roman-script languages—e.g. Hanukkah/Chanuka; Itzhak/Yitzhak; Petah-Tikva/Petach-Tiqwa, etc.

This is another issue entirely, and pertains to the deficiencies of quasi-phonetic spelling of Hebrew in Roman characters. For a more detailed explanation of this, and its possible solution in SimHebrew™, see my answer to What are the top countries with transliteration problem? —and follow the links.