Not just you, almost all the publishers will have this question when starting out. Google Ad Manager is an ad serving platform that can help you manage your ad inventories and sell them via multiple demand partners effectively.
AdSense is a demand partner (i.e., an ad network) itself. You can’t use Ad Manager to serve ads unless you are connected to a demand partner like AdSense. The most important aspect of Ad Manager is its ability to help you run both directly sold ads and non-guaranteed campaigns.
For example, let’s say you own a food website and you have partnered with a local restaurant to run ads. Typically, the advertiser wouldn’t buy all your impressions (say you have 1 million pageviews/mo. and 3 ads on a page, it leads to 3 million ad impressions in total and the advertiser may need just 1 million impressions). So, you have to sell the remaining impressions to other advertisers buying via demand partners.
Google Ad Manager makes it possible to do so. So, it isn’t the question of ‘or’, but the question of ‘when’. When you should start using Ad Manager and AdSense. If you have them both, it’s time for you to use them.