"To be honest, I often use wget to download files"
... some people tell me in a lowered voice, like if they were revealing one of their deepest family secrets to me. This is usually done with a slightly scared and a little ashamed look in their eyes - yet still intrigued, like it took some effort to say that straight in my face. How will I respond to that!?
I enjoy maintaining a notion that there is a "war" between curl and wget. Like the classics emacs vs vi or KDE vs GNOME. That we're like two rivals competing for some awesome prize and both teams are glaring at the other one and throwing the occasional insult over the wall at the competing team. Mostly because people believe it and I sort of like the image it projects in my brain. So I continue doing jokes about it when I can.
In reality though, where some of us spend our lives, there is no such war. There's no conflict or backstabbing going on. We're quite simply two open source projects busy doing our own things and we've both been doing it for almost two decades. I consider the current wget maintainer, Giuseppe, a friend and I'm friends with the two former maintainers as well.
We have more things in common than what separates us. We're like members of the fairly exclusive HTTP/FTP command line tool club that doesn't have that many members.
We don't have a lot of developer overlap, there are but a few occasional contributors sending patches to both projects and I'm one of them. We have some functional overlap in the curl tool with wget but really, I strongly recommend everyone to always use the best tool for the job and to use the tool they prefer. If wget does the job, use it. If it does the job better than curl, then switch to wget.
There's been a line in the curl FAQ since over 15 years: "Never, during curl's development, have we intended curl to replace wget or compete on its market." and it tells the truth. We are believers in the Unix philosophy that each tool does what it does best and you get your job done best by combining the right set of tools. In the curl project we make one command line tool and we make it as good as we can, but we still urge our users to use the best tool for the job even when that means not using our tool.
All this said, there are plenty of things, protocols and features that curl does that you cannot find in wget and that wget doesn't do. I've detailed some differences in my curl vs wget document. Some things that both can do are much easier to do with curl or offer you more control or power than in the wget counter part. Those are the things you should use curl for. Use the best tool for the job.
What takes the most effort in the curl project (and frankly that gets used by the largest amount of users in the world) is the making of the libcurl transfer library to which there is no alternative in the wget project. Writing a stable multi platform library with a sensible and solid API is much harder and lots of more work than writing a command line tool.
OK, I'll stop tip-toeing and answer the question you really wanted to know while enduring all this text up until this point:
When do you suggest I use wget instead of curl?
For me, wget is for recursive gets and for doing more persistent and patient retries of continuing transfers over really bad connections and networks better. But then you really must take my bias into account and ignore anything I say because I live and breath the curl life.