A "weight" is one of the parameters the model has learned, and the whole set of them is the model itself. If you can download them, the model is yours: you can run it without an internet connection, you can modify it, and nobody can take it away from you. If you cannot download them, you are the user of a service, however good that service happens to be.
This is not an ideological question but a very practical one about three things: where your data sits, what usage costs, and what happens when the vendor retires the model. The last of those gets mentioned least and turns out to be the most real: the big providers announce retirement dates on a regular basis, and a downloaded model file stays exactly what it was on that day too.
The most common misconception is that "open-weight" is the same as open source. It is not: the weights are available, the training data and the training process usually are not. The other misconception is that every open-weight model fits on your laptop — the best ones run to hundreds of billions of parameters and want serious hardware.
The three practical angles are worked through in Open-weight and Closed Models: What's the Difference for You?, a separate article. Directly related terms: local LLM, quantization and closed model.