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Some Feats, Traits, and other abilities are exclusive to worshipers of particular deities. And Clerics and other casters focused on a specific god are obviously effected mechanically by which.
That aside, without investing Feats, Traits, Class levels, or other resources, Deities do nothing for most characters mechanically. It's all flavor.
As for whether you can make your own, that's really up to your GM.
Generally speaking I'd suspect the answer is 'No' in most cases. Published settings tend to have a wealth of deities already, and adding more that your character worships is weird, potentially unbalancing, and makes little sense in the world. The same is true in the GM's own setting if they've put a lot of thought into it and have a deity list already. Some GMs might allow it anyway, but I'd expect them to be a minority.
Now, if you're doing a casual game in no particular setting, then I'd expect you could makeup your own God pretty easily. But those are rarer than games in existing settings in my experience.