When is it better to create STATISTICS instead of creating an Index?
You question revolves around - When is it a good thing to just create statistics vs create index (which create stats).

From my sql server internals notes (SQLSkills class- IE1 and IE2) and SQL Server internals book, below is my limited understanding :

SQL Server statistics are nothing but system objects that contain vital information about the index key values and regular column values.

SQL Server uses a cost based model to choose a "good enough" execution plan as fast as possible. Cardanility estimation (estimating no. of rows to be processed on each step of the 
query execution) is the most important factor in query optimization which inturn affects the join strategy, memory grant requirement, worker thread selection 
as well as choice of indexes when accessing data. 

SQL Server wont use nonclustered indexes when it estimates that a large no. of KEY or RID loopup operations will be required, so it maintains statistics on indexes (and on columns) which 
will help in such estimations.

There are 2 important things about stats :


Histogram stores info about data distribution for the leftmost statistics (index) column ONLY. It also stores info about multi column density of the key values. So essentially,
histogram stores data distribution for the leftmost statistics column only.
SQL Server will retain at most 200 steps in histogram irrespective of table size. The intervals covered by each histogram step increase as the table grows which leads to "less accurate" stats for large tables.

Remember that index selectivity is a metric which is inversely proportional to density i.e the more unique values a column has, the higher its selectivity.


When particular queries do not run very often, you can select to create column-level statistics rather than an index. Column-level statistics help
Query Optimizer find better execution plans, even though those execution plans are suboptimal due to the index
scans involved. At the same time, statistics do not add an overhead during data modification operations, and they
help avoid index maintenance. This approach works only for rarely executed queries.

Refer : 


Column Statistics Give the Optimizer an Edge by Kalen Delaney
Statistics, Damned Lies, and Statistics – What is Statman? by Conor Cunningham


Note: Someone like Paul White or Aaron Bertrand can chime in to provide more color to your good question.