A Hard Cost is an expense that you directly pay to a vendor (i.e court fees). You can enter in a hard cost from the New Check window.
When creating a Hard Cost you will reduce/credit your bank account and increase the selected account from the "Assign Accounts & Matters". If you'd like to bill your client for an expense, make sure that their matter is selected from the drop down and "Bill Client" is checked off. This expense will show when you invoice Unbilled Activities. Once the invoice is paid the original increase of the selected account will now decrease by the amount paid.
When creating a Soft Cost nothing will hit the books unless passed to your client by checking off the "Billable" option. This will add the cost to your Unbilled Activities. This entry will hit your books when the soft cost is invoiced on accrual basis and paid on cash basis. Once paid the your bank account will increase/debit and the "Soft Cost Revenue" account with increase/credit.