If the rent is paid at the beginning of the payment period (payments in advance, which is true for most leases), the rental payment pays the interest which was accrued and expensed the period before. For the first period of the lease, the entire payment is considered obligation (principal) repayment. During the period, interest is accrued and expensed, resulting in an accrued interest balance at the end of the period. On the first day of the next period, the accrued interest is paid, and the remaining rent reduces the outstanding obligation. The process repeats each payment period until the lease expires. In the last month of the lease, there is no interest expense, because the obligation is zero after the final rent payment is made. Many leasing systems do not recognize this timing difference, but this is correct accounting under FAS 13 (see FAS 13, Appendix C, Schedule 4, including the footnote to the 1/31 interest expense entry).
Whether the rent is paid at the beginning or the end of the period, interest expense and interest paid can also differ because interest expense is accrued each day, whereas interest paid is recognized only on the day of a rental payment.