Heavy performance drop on Progress query, looks like based on string character inside table

It all depends on the definitions and the key (and non-key) columns defined in the nonclustered index. The clustered index is the actual table data. Therefore it contains all of the data in the data pages, whereas the nonclustered index is only containing columns' data as defined in the index creation DDL.

Let's set up a test scenario:

use testdb;
go

if exists (select 1 from sys.tables where name = 'TestTable')
begin
    drop table TestTable;
end
create table dbo.TestTable
(
    id int identity(1, 1) not null
        constraint PK_TestTable_Id primary key clustered,
    some_int int not null,
    some_string char(128) not null,
    some_bigint bigint not null
);
go

create unique index IX_TestTable_SomeInt
on dbo.TestTable(some_int);
go

declare @i int;
set @i = 0;

while @i < 1000
begin
    insert into dbo.TestTable(some_int, some_string, some_bigint)
    values(@i, 'hello', @i * 1000);

    set @i = @i + 1;
end

So we've got a table loaded with 1000 rows, and a clustered index (PK_TestTable_Id) and a nonclustered index (IX_TestTable_SomeInt). As you've seen in your testing, but just for thoroughness:

set statistics io on;
set statistics time on;

select some_int
from dbo.TestTable -- with(index(PK_TestTable_Id));

set statistics io off;
set statistics time off;
-- nonclustered index scan (IX_TestTable_SomeInt)
-- logical reads: 4

Here we have a nonclustered index scan on the IX_TestTable_SomeInt index. We have 4 logical reads for this operation. Now let's force the clustered index to be used.

set statistics io on;
set statistics time on;

select some_int
from dbo.TestTable with(index(PK_TestTable_Id));

set statistics io off;
set statistics time off;
-- clustered index scan (PK_TestTable_Id)
-- logical reads: 22

Here with the clustered index scan we have 22 logical reads. Why? Here's why. It all matters on how many pages that SQL Server has to read in order to grab the entire result set. Get the average row count per page:

select
    object_name(i.object_id) as object_name,
    i.name as index_name,
    i.type_desc,
    ips.page_count,
    ips.record_count,
    ips.record_count / ips.page_count as avg_rows_per_page
from sys.dm_db_index_physical_stats(db_id(), object_id('dbo.TestTable'), null, null, 'detailed') ips
inner join sys.indexes i
on ips.object_id = i.object_id
and ips.index_id = i.index_id
where ips.index_level = 0;

Take a look at my result set of the above query:

As we can see here, there are an average of 50 rows per page on the leaf pages for the clustered index, and an average of 500 rows per page on the leaf pages for the nonclustered index. Therefore, in order to satisfy the query more pages need to be read from the clustered index.

Without full-text search, no, there's no magic to making string parsing faster within SQL Server, other than pre-calculating results or throwing more resources at the problem.

If you have a narrow set of search patterns that are repeated over and over again, it's possible you could maintain a skinnier materialized portion of the table that meet those criteria (e.g. a table of just the PK columns representing the rows in the main table that match '%ABC%' - you could maintain these through triggers). This will reduce the amount of reads required, but may not have a serious impact on duration.

If people are entering arbitrary search strings in a non-repeatable and unpredictable way, that may not help anyway.

10 seconds seems like a long time for a table with 160K rows. If you are on V12 (and can run this query in relative isolation), you should be able to determine the waits that changed during that query, using sys.dm_db_wait_stats - it may be that you can't keep 300MB of data in memory and the wait time is all disk churn. In this case it may just be that you are sharing an overwhelmed server, so one consideration would be to move up to a better tier that provides for better performance.

Another option you can consider is application-side caching (think memcached, redis, etc.), where you have a copy of the data in your application's memory, and perform the searching there instead of within SQL Server.