Networking is entering a scenario with a selfish intention toward a specific outcome, and forcing an interaction to that end. Spot someone who can help you, small talk for 5 minutes, ask for a favor.
Connecting is entering a scenario with a sense of wonder. You’re expecting to meet interesting people and that alone is exciting. On top of that, you have value to offer each of them. Favors, introductions, recommendations.
The former is pure awkwardness in pursuit of a short term reward. And it’s just as gross via DM as it is in person.
The latter is confidence personified. The rewards are never exactly what you think they might be, but they’re bigger, better, and they come later.