querying binary column using like in sql server

Don't convert it, but treat is as a range (like you would datetime values)

DECLARE @foo TABLE (TestData varbinary(100) NOT NULL);
INSERT @foo (TestData) VALUES
         (0x0001DC780C0030373156635D0C00B8840301009A0600AC),
         (0x0001AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA),
         (0x0001AFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF),
         (0x0301DC780C0030373156385D0C006499C401009A0600AC),
         (0x0301FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF),
         (0x0302000000000000000000000000000000000000000000);

SELECT *
FROM @foo
WHERE TestData >= 0x0001 AND TestData < 0x0002;

-- added more digit for clarity of what actually happens
SELECT *
FROM @foo
WHERE TestData >= 0x00010000 AND TestData < 0x00020000;

SELECT *
FROM @foo
WHERE TestData >= 0x0001AA AND TestData < 0x0001AB;

SELECT *
FROM @foo
WHERE TestData >= 0x0301 AND TestData < 0x0302;

This has the bonus of being able to use an index on TestData

Edit, you just specify as many digits as you need

For a leading prefix LIKE comparison, gbn's answer will do. For a real LIKE equivalence of string searches, you can use LIKE as follows:
(borrowing schema and sample data from @gbn)

DECLARE @foo TABLE (TestData varbinary(100) NOT NULL);
INSERT @foo (TestData) VALUES
         (0x0001DC780C0030373156635D0C00B8),
         (0x0001AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA),
         (0x0001AFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF),
         (0x0301DC780C0030373156385D0C006499C401009A0600AC),
         (0x0301FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF),
         (0x0302000000000000000000000000000000000000000000);

SELECT *
FROM @foo
WHERE CAST(TestData AS VARCHAR(MAX)) LIKE '%'+CAST(0xDC78 AS VARCHAR(MAX))+'%';

When you cast a binary value to VARCHAR, all it does is treat the raw bits as a string stream. It does not magically convert it into the string representation. Consider the example below:

select cast(0x41 as varchar(10));     -- Result: A
select cast(0x414263 as varchar(10)); -- Result: ABc

Because the byte 0x41 or ordinal 65 is 'A' in the standard Latin codepage.