HAProxy vs. Nginx
Solution 1:
haproxy is a "load balancer" it doesn't know to serve files or dynamic content. nginx is a web server capable of many interesting things. if you only need to load balance + HA some third web server then haproxy is enough. if you need to implement some static content or some logic in routing of the requests before terminating them on a third server then you may need nginx.
The reason you can see haproxy+nginx on the same host is that it allows you to bring down single nginx instances while haproxy continues to serve requests from other hosts. Imagine having a RR DNS using A records:
myapp.com IN A 1.1.1.1
myapp.com IN A 1.1.1.2
where 1.1.1.1 and 1.1.1.2 are two hosts with haproxy+nginx configured to load balance between them. Now for some reason your 1.1.1.1's nginx goes down. The browsers that come to 1.1.1.1 are still being served by haproxy on it which in turn gets data from 1.1.1.2's nginx.
hope it helps
Solution 2:
HAProxy is definitely the better, more fully featured loadbalancer (compared to the free nginx, not nginx plus (but one could argue that as well).
One thing that HAProxy sadly still can't do is generic UDP connections. So we used HAProxy and nginx on our logging lbs. But HAProxy released support for syslog/udp in 2.3 so we are about to change that. :)
Solution 3:
We use HAProxy together with nginx. There are a number of reasons.
Nginx can do everything (more or less) but you don't want your load balancer serving web pages. Some error in config (which might have nothing to do with load balancing) and your entire setup comes to a screeching halt. Imagine that you have a Nodejs app, a Dotnet Core app, static files served by Nginx, and a php app. You just make some mistake and your 4 apps come to a standstill. You have lost your redundancy too if you have multiple instances of each app.
Even if you say that Nginx will only do the load balancing, Nginx doesn't support PROXY Protocol which is problematic if you forward to other servers who are also not serving the pages.
In addition there is something to be said for doing one thing and doing it well. Nginx is the master toolbox today. It does almost everything. Your load balancer is supposed to be the most stable part of your setup. Wouldn't you prefer to use something that was built just for load balancing?
If you use varnish then HAProxy works well with it and in fact they are made by the same people.
If you want an added level of balance then you can also use dns as a load balancer with multiple HAPROXY instances. Dns is not meant for this perse but you will always have some weak link. Your load balancer can crash too even if it's managed by your cloud provider. Most web browsers today will try other servers if there is more than one in your dns entry so it's like a load balancer. Your dns should be very reliable thus increasing your uptime.
We use 2 haproxy instances with 2 varnish instances with two dns entries.