Excel ran out of resources while attempting to calculate one or more formulas

The reason is probably differing 32-bit memory fragmentation on the 2 Pcs (its often difficult to use all of the theoretically available 2 GB).
You can reduce the amount of memory used by the pivot table by reducing the number of columns and/or reducing the memory required by some of the columns (text strings are a good candidate).
(You can measure the amount of memory used by the pivot cache with VBA PivotCache.memoryUsed)
I assume you are creating the pivot cache by reading the data from the query directly into the pivot cache rather than putting the query data onto a worksheet and basing the the pivot on the worksheet, which would use much more memory.
You don't say what the >1000000 formulas are so I don't have any suggestions for improvement for them.
If you want to use large amounts of data in Excel pivots etc you would get better results from the 64-bit version of Excel 2010 which does not have the 2-GB limit

thanks for all the suggestion, readings and help. I have solved the problem by

1. Change all formulas into values step by step per ~ 100,000 rows --- This reduce the memory held by excel by 10%
2. Removed any unnecessary data/Worksheet(s) in the workbook (The raw data before data massaging ) --- This reduce the memory held by excel by 40%

And the reading http://www.add-ins.com/support/out-of-memory-or-not-enough-resource-problem-with-microsoft-excel.htm suggest why 6G RAM PC is facing this issue rather than 2G RAM PC. I think it's because the 6G PC has a lot of excel add-in on it consuming RAM as well.

Thanks for the helps.