So that you can store/retrieve data without knowing the schema.
Oh but you have to know the schema right? Yes, some other part of the application knows the schema, but this part doesn't have authority over the DB. Also, the schema may be data as well.
Well, it's slower because of a join that exists in SQL (the relationship between your 'field/value' table and the entry. Apart from that, as you said indexes are similar.
The fun thing about NoSQL skeptics is how they think of only the current scenarios they work with, and they won't believe you until they get burned by it. So be it.
Oh but you have to know the schema right? Yes, some other part of the application knows the schema, but this part doesn't have authority over the DB. Also, the schema may be data as well.
NoSQL reduces the work needed for that.