You've probably worn the cogs on the old cassette, most likely from using your chains too long before replacment The bushings in the chain will wear after 500 to 2,000 miles changing the distance from link to link, (stretching). When this happens the links put more pressure on fewer and fewer links because the contact points are no longer fully engaged, wearing the cogs. Your favorite gears are always the ones to go first because you use them more. The wear can be noted by either a slightly widened area were the chain contacts the cog under pressure and by the cog taking a "saw tooth" or "sharkfin" shape. Park tools makes some gauges to measure your cogs and chains for wear so you can replace them before they're worn enough to cause damage, you can measure the chain with a ruler , but it is a pain in the a** to do. If you stay on top of the chain measurement/wear the cogs will last for a very long time. If you go for the gauges get the chain gauge first, for most people it's the most useful.

You chain-rings can/will suffer the same damage as the cogs but luckily not as often as there are more of them to spread the load