Linguistic Drift vs. Deity Drift

The simplified relation between the languages ...
The simplified relation between the languages Dutch, English and German. (Photo credit: Wikipedia)

Languages evolve, and populations that once spoke the same language will end up speaking different (although related) languages given enough time, if those populations become distinct. For example, English is a Germanic language, but an English-only speaker and a German-only speaker couldn’t converse with any understanding. The English word for the stuff in oceans, lakes, and streams is “water”; in German it’s “Wasser”. Different words meaning the same thing.

So what about deities? Are (for example) Odin and Wotan the same god, and only the name changes, depending on the people? Or are they different, if similar, gods? How does one make that determination?

Invariably one’s understanding of the nature of deity comes into play in this. It’s relatively easy for soft polytheists like myself to answer this question, but I’m interested in what hard polytheists have to say too.