How do AMP and PWA differ from each other?

Both actually are complementary to each other but differ in how the implementation works.

PWA, while it helps create a web app, still makes pages faster for a user. Basically, every site makes a lot of network calls.  There is a lot of network requests and traffic to show styles, scripts, images, etc, elements to be downloaded, page views, and on average it takes

200-300ms per request to the customer. What our app PWA app does is, after the first initial load we find the resources that are the same on every page (for example, logo, styles, fonts, scripts, etc) and cache them, so when customers open the next page it's much faster. A lot of these network requests do not need internet and are opened right away (0ms). This makes the site much faster, reduces the data usage a lot, and allows the site to even work offline. However, PWA speed increase tends to favor repeated users of the site rather than new users.  Importantly, your site remains the same and does not change at all in the look and feel. 

AMP tends to work differently. It helps create separate AMP pages which are optimized for speed (by placing restrictions and rules on the contents and formats on these pages that decrease the speed). In addition, these pages are cached in servers close to where users are located. Every time a user makes a request, AMP gives a speed boost because 1) the AMP page design naturally makes them faster and eliminates sources of delay or tells the browser how to optimize the view 2) the content is delivered from servers close to the users minimizing latency. However, because of the constraints in the AMP framework, the look and feel may differ from your original web page. We have tried to replicate to a high extent the look and feel based on popular themes but while the AMP page is fast, the web page looks a bit different. 

Hence both solutions are useful from a speed perspective but differ a bit in how they work. PWA pages help a lot in speed when you have returning users and not so many new users. AMP pages help both new and returning users. They both tend to improve significantly the SEO rank. On PWA, the look and feel of your site remain exactly the same while on AMP, the look and feel changes and you have to try to keep the AMP page look consistent with your non-AMP pages. I hope this helps you understand the options better. 

We see many of our clients use AMP product and collection pages, etc. in conjunction with their PWA. I am sure you will see a benefit by using PWA in general and by AMP in optimizing your landing pages.