Omg, thanks so much!!!! Your explanation rocks and I understand better than ever!!!!!I'd say 설득시키다 is when you persuade someone to agree on your ideas (so the person you persuaded will be on your side and do/say pretty much the same things, while 납득시키다 simply means you have someone listen to you and understand the way you pursue)
For example, you would persuade him/her if you want to get married to the same-gender partner. So you guys get married if you got a yes from the partner after persuading him/her, and then you guys be on the same side. That would happen when you 설득시켰다 someone. However, those who listened to you carefully and ended up understanding your lifestyle actually don't have to have the same-sex marriage, but they could just support you and want you (two) to live happier when you 납득시켰다 someone. You can also think of 납득시키다 as you need to 납득시키다 first in order to 설득시키다 someone.