There's no difference. But везде is more common. You can also say повсюду (the meaning is the same)

I feel absolutely no difference in their meaning

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.

Всюду" is outdated word enough and a little bit country. I hear this word really rarely. You had better use "Везде".