This worked perfectly for me in Google Spreadsheets / Google Documents / Google Docs
=(SUMIF(A:A,">= 01/12/13", E:E))-(SUMIF(A:A,">= 31/12/13", E:E))
If you want to reference cells in place of writing the date '01/12/13' in the formula itself, use the logical operation in quotes, then an &(ampersand) and the cell reference: ">="&B:1
=(SUMIF(A:A,">="&B:1, E:E))-(SUMIF(A:A,">="&B:2, E:E))
Also the > (greater than) or < (less than) sign must come before the = (equals) sign.
The above formula works like so: all values after 01/01/13 = 350. All values after 31/01/13 = 250. Therefore values between 01/01/13 and 31/01/13 = 350 - 250 = 100.
Also here is the link to the SUMIF Google Help Page
I hope that's all clear.