Although the adjectives illegible and unreadable both refer to texts that can’t be read, their conventional uses are different.
Illegible refers to texts that can’t be read due to bad handwriting, physical deterioration, or any other issue that makes the words difficult to decipher. So these writers use illegible in the conventional sense:
Have you ever noticed that ‘creative types’ seem to have a similar handwriting; doctors, their illegible scrawl; mathematicians, their small and steady characters? [Creative Boom]
It is printed on a single sheet of foolscap, and the writing is so small that it’s illegible. [New Yorker]
While texts that are illegible are technically unreadable, unreadable is used mainly in reference to texts that are dull, nonsensical, uninteresting, or difficult—for example:
Dylan’s first book, the best forgotten Tarantula from 1966, was an unreadable amphetamine-fuelled stream of consciousness blank poetic opus, after all. [Telegraph]
The grammar isn’t actually wrong, but the stuttering created by the four commas makes this almost unreadable. [NY Times]
This distinction is not a rule, however, and the words are sometimes used interchangeably. These writers, for instance, use unreadable where illegible might be more conventional:
Every time I print a screen, it comes out in a large font and the letters run on top of each other, making the text unreadable. [IT Jungle]
A poem honoring those who have lost their lives at sea on this pillar at Memorial Park Cemetery in Unalaska is nearly unreadable. [The Dutch Harbor Fisherman]