Both words mean blind. I tried doing some research but I couldn't find the difference.
盲目の is close to "blind". It is not derogatory IMHO, but I must say it is often sensitive because a few people dislike the 亡 ("death") component of 盲, just as there are people who dislike 障害. Although it is still used in solemn literary contexts today, it is no longer a primary choice.
目の不自由な is more like "visually-challenged". It sounds much politer and politically correct. In addition, 目の不自由な方 also includes those with low vision as well as totally blind people.
Another important phrase is 視覚障害のある ("with visual impairment") or 視覚障害者 ("visually impaired person"). I think this is the most neutral and safest phrase today.
盲目 is a more clinical term that means "blindness." 目の不自由な is a more euphemistic, warmer phrase that you often used when talking about blind people, especially in a non-clinical setting. English analogues might be someone who is "optically challenged" or "vision impaired." Same meaning, but a bit more empathetic.