Why are multiple COUNT faster than one SUM with CASE?
The COUNT(*) version is able to simply seek into the index you have on the status column once for each status you are selecting, whereas the SUM(...) version needs to seek the index twelve times (the total number of unique status types).

Clearly seeking an index three times is going to be faster than seeking it 12 times.

The first plan requires a memory grant of 238MB, whereas the second plan requires a memory grant of 650MB.  It may be that the larger memory grant could not be immediately filled, making the query that much slower. 

Alter the second query to be:

SELECT  Approved = SUM(CASE WHEN Status = 'Approved' THEN 1 ELSE 0 END),
        Valid    = SUM(CASE WHEN Status = 'Valid'    THEN 1 ELSE 0 END),
        Reject   = SUM(CASE WHEN Status = 'Reject'   THEN 1 ELSE 0 END)
FROM dbo.Claims c
WHERE c.Status = 'Approved'
    OR c.Status = 'Valid'
    OR c.Status = 'Reject';


This will allow the query optimizer to eliminate 75% of the index seeks, and should result in both a lower required memory grant, lower I/O requirements, and faster time-to-result.

The SUM(CASE WHEN ...) construct essentially prevents the query optimizer from pushing the Status predicates down into the index seek portion of the plan.