Hard question, because both have their benefits and issues. I love doing both - it's just depends on the mood I'm in when picking up a drama. ...I could talk about this all day.
I prefer marathoning a completed drama when I need a full story just thrown in my face, as it's the most able to do that. You get the full series without waiting and without having any real issues of it. So you can go through it non-stop, and you can pick it up whenever you want happily, knowing the next episodes are there waiting for you. 2am and want to watch it? No problem, the subs are there for the people who don't know the language and it's already aired - so no worries about there being no episode there. Secondly, people already know whether the full series is good or bad in reviews, which means you have a more accurate view of whether you will like the series or not. Not saying reviews will be 100% accurate mind you, people already having fully formed opinions on the series is also an issue as well as a benefit, because some will say it's good, some will say it's bad. It's mixed, and you can have no idea which is correct. It can be just as much as a leap game as an airing one.
The bigger downsides I've found are that you can complete it at speed which means it will be gone quicker and you'll have to find something else that replaces it - which is a bummer if you really liked the series. There's also sacrificing your sleep - especially without realising. ("Oh God, it's now 4am. When, and how?") It can be horrendous the next day, especially if you have something important to do.
Nevertheless, I like ongoing for the excitement factor and for it's schedule format. For ongoing you get an episode (or a couple of episodes, depending on the place it comes from) every week. You're not going to have any worries for a good amount of time about having a drama to watch because it's airing on a schedule for you and you have a full week to be excited about it, and make up your own theories as to what will happen until then. It has current people in, so you're learning more about the current industry, and you get to see the raw versions of the episodes (which is kind of cool) which I don't see frequently with a completed series as it's already subbed. On top of it, you get to talk to people about your current opinions and see how everyone's reaction changes depending on how the series develops. That, for me, is the most interesting part to watch.
The downsides are bigger though, in my opinion, especially if you don't have any patience in you. You are restricted to the amount of episodes you're given, schedules can change unexpectedly (due to events in the place the drama is airing) and you're waiting for that next episode, next week instead of this week. There's also waiting for the subtitles to be done if you don't know the language, which can end up problematic too if they stop subbing for unexpected reasons. And lastly the spoilers - ugh. You get spoilers with marathons, but the ongoing ones are critical hits instead of just "spoilers" because everyone is talking about it. There are spoilers everywhere waiting to creep up and spoiler you. The only way you can avoid them is staying away from the internet, and that is such a big sacrifice.
(Spoiler talk: Reply 1997 was my worst hit - I found out who she got with, an episode before she got with him, after waiting all that time. I was so annoyed. I already knew who she was going to get with before the spoiler, but that's not the point. Darn you spoilers!)
And that is the longest post I have probably ever posted on MDL. ...Whoopsie.