@vilas, the 320 series uses the CFM56-5 series whereas the 737NG uses the more modern CFM56-7 series. So no, they are not the same engine. The last variant for the NG (CFM56-7BE) was only introduced in 2010 and among other things it brought a change to the inflight start envelope and therefore relighting with a loss of thrust on both engines.
The checklist calls for 275kts above FL270 and 300kts below, however as others have pointed out that is not meant to exceed MMO.