Pivoting Multiple Columns Based On a Single Column

You can get the desired result but since you need to PIVOT on multiple columns (TaskDate1 and TaskDate2) I would first unpivot the two columns into multiple rows.

Since you are using SQL Server 2008+, you can use either the UNPIVOT function or CROSS APPLY to convert the multiple columns. The basic syntax will be:

select projectid,
  col = TaskType+col,
  value
from yourtable
cross apply
(
  values 
    ('Date1', TaskDate1),
    ('Date2', TaskDate2)
) c(col, value)

See SQL Fiddle with Demo. This will convert your data into an easy to use format:

| PROJECTID |        COL |                        VALUE |
|-----------|------------|------------------------------|
|         1 | Type4Date1 | April, 01 2013 00:00:00+0000 |
|         1 | Type4Date2 |   May, 06 2013 00:00:00+0000 |
|         1 | Type0Date1 | April, 12 2013 00:00:00+0000 |
|         1 | Type0Date2 |   May, 08 2013 00:00:00+0000 |

Once you have the data in this format, then you can apply the PIVOT function to the get the max(value) for each col:

select projectId,
  Type0Date1, Type0Date2, 
  Type2Date1, Type2Date2,
  Type4Date1, Type4Date2
from
(
  select projectid,
    col = TaskType+col,
    value
  from yourtable
  cross apply
  (
    values 
      ('Date1', TaskDate1),
      ('Date2', TaskDate2)
  ) c(col, value)
) d
pivot
(
  max(value)
  for col in (Type0Date1, Type0Date2, Type2Date1, Type2Date2,
              Type4Date1, Type4Date2)
) piv;

See SQL Fiddle with Demo.

The above version will work great if you have a limited number of values per ProjectId, but if you have an unknown number of Tasks for each project, then you will need to use dynamic SQL to get the final result:

DECLARE @cols AS NVARCHAR(MAX),
    @query  AS NVARCHAR(MAX)

select @cols = STUFF((SELECT ',' + QUOTENAME(TaskType+col) 
                    from yourtable
                    cross apply
                    (
                      select 'Date1', 1 union all
                      select 'Date2', 2
                    ) c(col, so)
                    group by col, tasktype, so
                    order by tasktype, so
            FOR XML PATH(''), TYPE
            ).value('.', 'NVARCHAR(MAX)') 
        ,1,1,'')

set @query = 'SELECT projectid, ' + @cols + ' 
            from 
            (
              select projectid,
                col = TaskType+col,
                value
              from yourtable
              cross apply
              (
                values 
                  (''Date1'', TaskDate1),
                  (''Date2'', TaskDate2)
              ) c(col, value)
            ) x
            pivot 
            (
                max(value)
                for col in (' + @cols + ')
            ) p '

execute sp_executesql @query

See SQL Fiddle with Demo. These will give a result:

| PROJECTID |                   TYPE0DATE1 |                 TYPE0DATE2 |                   TYPE2DATE1 |                 TYPE2DATE2 |                   TYPE4DATE1 |                 TYPE4DATE2 |
|-----------|------------------------------|----------------------------|------------------------------|----------------------------|------------------------------|----------------------------|
|         1 | April, 12 2013 00:00:00+0000 | May, 08 2013 00:00:00+0000 | April, 20 2013 00:00:00+0000 | May, 17 2013 00:00:00+0000 | April, 01 2013 00:00:00+0000 | May, 06 2013 00:00:00+0000 |