There is no difference for real between these two words.
They can be both used.
Везде is more informal and common word nowadays.
Всюду sounds more poetical, I would say...
We also have some phase, it sounds like "везде и всюду" - example: Я искал её везде и всюду.
See, these two ones are used at same time, but they mean the same thing ("everywhere"). Here, they are used for strengthening, i.e. he was looking for her everywhere really.