- Huge HP and high Special Defense make it the best special wall around, allowing it to check things like Volcarona and Specs Kyurem.
- Great supporting movepool with access to Heal Bell, Wish passing, Toxic/Thunder Wave for status, and Teleport.
- Its ability, Natural Cure, lets it absorb status.
- Only one weakness and an immunity to ghost.
- It relies on Seismic Toss to deal damage, meaning it can be easily be set up on by Ghost types, especially those with Substitute or Taunt.
- Easily overwhelmed by physical attackers and moves like Psyshock due to low defense.
- Can be practically rendered useless when tricked a Choice item.
- Competes for usage with Chansey (more on that below)
- No resistances.
Despite these cons, Blissey's ability to take special attacks is unmatched, and the variety of supporting moves and utility it has makes it a great pick that can fit easily on a lot of teams.
Blissey vs. Chansey
I know this wasn't a part of your question, but I think its definitely worth talking about. Since Eviolite was introduced in Gen 5, Chansey was usually regarded as much better than its evolution, but I think the two are as close in viability now as they have ever been.
Since they both do practically the same thing, the held item is the key here. It's true that Chansey is bulkier with Eviolite, but Knock Off has become very common to see in this generation. Luckily for Blissey, it doesn't require its held item to soak up damage. Leftovers is nice for obvious reasons, but I think Heavy Duty Boots is actually a great item for Blissey. Being able to switch in without taking any damage from entry hazards is something Chansey can't do no matter what. It pairs nicely with being able to Teleport pivot right back out. I recommend trying it.
I think stallier teams would prefer to have that extra bulk that Chansey provides, however I think Blissey fits even better on balanced teams that don't necessarily need that and would prefer the extra utility Blissey provides instead.
Although I wouldn't recommend using any attacking move on it other that Seismic Toss, it might also be worth noting that Blissey has a usable Special Attack stat compared to Chansey's, at least.