Main difference: Instead of ETFs, you buy the different crypto coins.
The similarities are actually extensive:
1. People go to Coinbase mainly to invest their money. Sure, maybe, one day, in the distant future, perhaps, coins will be used for commercial purposes; but now and in the foreseeable future they are investment (speculation) vehicles
2. Vanguard has a direct relationship with consumers; you can open an account, wire money, and buy Vanguard funds. Same with Coinbase (and not the case with stock exchanges as some have suggested)
3. There is a certain amount of trust in the brand that makes people want to buy the funds/assets or wire their money to these brands. But that has limited power (see point below)
4. Vanguard's products are commodities, just like crypto coins are (you can buy the same bitcoin in many places, and you can buy essentially the same S&P 500 ETF from many platforms)
The difference is that Vanguard is successful thanks to a focus on low cost funds; Coinbase still rides first movers advantage. But inevitably it'll have to compete on cost.
Vanguard is managing $6+ T of actual assets; that's many times the total market cap of all cryptos.
And now here's the question: if they had the same valuation, would you put your money on Vanguard or Coinbase?