Well, refrigerators have gotten significantly more energy efficient over the years (when compared apples-to-apples with models with similar features*). So you might be missing out on some electricity for using an older one. And since a fridge is an appliance that runs at all times, energy efficiency can be especially important -- more important than most other appliances, which are only used occasionally.
*However, some older fridges can be more efficient than some newer fridges. Design and features play a role. Two door with the freezer on top is the most efficient design, however a lot of modern fridges tend toward multi-door designs with the freezer on the bottom, which is less efficient at keeping cold in, even if the compressor is more efficient at making it cold. Also, in-door ice makers are very bad for efficiency, and almost ubiquitous on modern fridges.
That said, though, you can still buy modern 'dumb' fridges with a simple, efficient design, no 'smart' features, and maybe even no ice maker in the door. That's the best choice for energy efficiency. (After all, running the screen and the smart features also uses electricity.)