These two offerings seem on the surface to do the same thing, but in practice they’re completely different tools.
Which you should choose right now depends on your goals. Most startups these days run on AWS. Amazon has a zillion services under the AWS banner that handle all kinds of needs. Many of them are better than others.
Digital Ocean, on the other hand, does one thing and does it well. Bargain web hosting. They have a much leaner feature set and so you often have to implement something yourself that Amazon has a product for.
The downside to Amazon isn’t so much in the added cost, but in the lock-in effect. Once you build your product on top of Amazon, it’s extremely difficult to move off the platform. Amazon is also sufficiently complicated that you need to specialize in it to be able to work effectively with it.
Whereas with Digital Ocean you can get started with the skills you already have, you generally don’t have to learn anything Digital Ocean-specific to get things done.
If you’re looking to, at some point, get a job in the startup world, pick AWS. You’ll be more employable and be able to onboard onto a new job more quickly where they use AWS. It’s a harsher learning curve but the eventual career benefits will make it worth it.
If you’re looking to build out your own product, go with Digital Ocean. It’s simpler and friendlier.