They're pretty close at first glance. Using Affinity's apps won't be difficult for you beyond learning what the different tools are called, where they are, and what palettes are available, etc. But the deeper you dive into the apps, the less capable they are compared to the Adobe apps. This is perfectly understandable considering the maturity of Adobe's apps and the development behind them, but worth noting.
For most users, they're a perfectly acceptable replacement for Photoshop and Illustrator for individual users or small in-house design shops that handle everything internally. If you work with others who use PS and AI, or you're a power user, the waters are greatly muddied.
The level at which I use Photoshop, Affinity can't even compete - it's just not even worth launching the app. For instance, Affinity Photo doesn't support Photoshop's Smart Objects... that's a deal breaker right off the bat for me. There are dozens upon dozens of other features Affnity Photo doesn't have that I use daily.
Illustrator is another ball game though. My use of Illustrator is mostly simple logo design and fairly simple illustration editing, so I could very easily use Affinity Designer as a replacement.
What I tell people who ask the same question as you is this: If you don't work with other people's files and you don't need to send your editable files to other people, then Affinity Photo, Designer, and Publisher are fantastic. But if you expect to share your files and use other people's files, then don't waste your money on the Affinity apps because it won't be long before you're hitting roadblocks.
As far as Affinity Publisher as an Adobe InDesign replacement... don't make me laugh. It's barely more useful than Apple Pages in that regard, and a lot more convoluted. But it doesn't sound like you need or use InDesign anyway.
