The difference between play and game is especially hard for French speakers (as I am) as we use the same word for both aspects: "le jeu".
When I have to explain the distinction in French, I express it like that:
Play is non-structured moments of fun for the sake of pleasure
Game: structured fun with rules, goals, end results, winners and losers
Playful before gamified
In some way, we could imagine that we add to every service a touch of playfulness, but forcing gamification in every service could make life a bit overwhelming.
What others say
"Make something playful (not gamified) is a great design principle in general. To make something playful you have to have some thought about the person who will be using it and it always feels lovely when, as a user/customer, you realise someone has thought about the moment for you."
— Andy Polaine in a comment in a story of playful service
Examples of playful services
Back in the day, Volkswagen launched a campaign called Fun Theory that showed that by adding just a little bit of playfulness, we can make boring actions that nobody wants to take (like taking the stairs, or throwing stuff in the bin) more enjoyable to do.
For example, by adding a playful sound to a trash bin, the team was able to collect in one day 41 more kg of trash in one trash bin compared to a classical trash bin.