U2 has an album named "Songs of Innocence" and a second, more recent album named "Songs of Experience". These appear to at least be named after the William Blake poetry. What is the relationship between the CDs and the William Blake poetry? How did the poetry inspire the CDs and song lyrics?
The resemblance is mostly thematic.
According to Bono (lead vocalist and primary lyricist of U2), the main thing they took from Blake was the idea of comparing innocence and experience:
I try not to talk about William Blake too much because it sounds pretentious quoting such a literary giant but it was his great idea I pinched to compare the person we become through experience to the person who set out on the journey. If you're talking about innocence, you've probably already lost it but I do believe at the far end of experience, it's possible to recover it with wisdom. I'm not saying I have much of that but what little I have, I wanted to cram into these songs.
An analysis by Tassoula E. Kokkoris, probably the closest thing there is to an 'expert on U2', goes into more detail about similarities in theme and ideas between Blake's Songs of Innocence and Experience and the eponymous U2 songs. This was written before the release of U2's Songs of Experience, but it's pretty insightful nonetheless. Do read it in full, as it backs up its arguments with relevant quotes from both Blake poems and U2 songs, but I'll quote the most important parts here.
In obvious terms, the band took the name for their current (and presumably next) album from [Blake's] poetry books. In a broad sense, the themes Blake explores in his texts (childhood innocence, social injustice, poverty, conflicting aspects of religion) are identical to those of U2.
Bono is quoted as saying the Songs Of Innocence album is all about "first journeys" and "falling in love." We can only assume that the second volume will reflect an older perspective.
The fact that Blake released these works of art in a new and inventive way for his time is also a parallel, as (love it or hate it) U2 released their album via iTunes in a way no other band had ever before. [...]
Even the way the songs are presented - heavy on the visuals, with books and pages falling from the stage and sky, respectively, each night - mimics the spirit of Blake's ancient copperplates; conveying images in a way that had never before been seen. No other lead singers have leapt into a cage mid-show that displayed their childhood street as they sang along to the description of it. Nor have concertgoers clamored to collect ripped pages of books they were (most likely) forced to read in young adulthood, searching for the symbolism as they Google paragraphs of old text on their smartphones.
Kokkoris notes that both the Blake poems and the U2 songs have an underlying theme of appreciation of nature:
An undeniable infusion of nature is repeated throughout William Blake's collection. Whether he's "weeping in the evening dew" or has "smil'd among the winters snow", he's embracing the natural world. [...] Similarly, U2 lyrics on Songs Of Innocence are rich with landscapes and environmental descriptions.
Nature is also used to describe a memorable childhood place. In Blake's The Ecchoing Green, it's a location where kids played in their youth [...] For Bono, the beauty found in his best friend's yard brings comfort to "Cedarwood Road"
There are countless other references to nature in both works that mention the ocean, the sky, stars and seasons. A relatable and universal way for each artist to convey a mood - even if over 200 years apart in delivery.
She also compares their attitudes towards religion. Both Blake's poems and the U2 songs display a positive portrayal of God but a more negative portrayal of the church.
Though neither Blake's poems nor U2's album could be considered strictly religious, they're both laced with references to Christianity. God and the church are both celebrated and condemned in each collection.
In Blake's poem The Divine Image, he urges non-judgment and shows that God is compassionate [...] In "Lucifer's Hands", U2's singer delights in God "saving" him through music
Alternately, Blake takes God to task for the suffering of the poor in Holy Thursday [...] And in U2's "Sleep Like a Baby Tonight", shades of priestly abuse are evident
In Blake's time, he was considered radical for his vocal opposition to the Church of England. While Bono, The Edge, Larry and Adam are probably known more for their political voice, all but Adam have previously identified as religious, so it's a slight about-face to see them so blatantly criticize the church.
[Ra'T note: Blake's Songs of Innocence and Experience are definitely religious. Many of them are very explicit in praising God, who even appears in person in some of them. What they don't do is consistently praise established religion, in the form of the Church, but it's not really cognitive dissonance to both love God and hate the Church establishment.]