Is there any reason you should buy a router to replace the one your ISP provided? I'm on my third one now, all provided for free and all worked with no issues. But would I gain something choosing a branded one? Privacy, reliability, included software...? And would there be compatibility problems? Do you have to call one of their technicians and work out bits and bobs? Thanks.
It depends. I am using my ISP provided router for last 8 years (meantime it survived a lightening that fried my mobo), and though its showing its age now, most of the time it worked flawlessly. One of the big advantage for a router purchase outside of your ISP is you get a lot of choices. Most of the ISPs would provide you only 4-5 models to choose. And if you choose on your own you can get one with sufficient wireless strength and so.
To replace the hardware usually gives you more control and choice. Generally the ISP stuff is dated by at least 5 years to what passes for new on Newegg. Of course, depending who your ISP is, you might not be able to easily swap out the hardware. Or if you can, then they'll usually want you to pick through a list of "officially supported" equipment (which still, some of what's on their list can still be decent. Judge by reviews elsewhere online). Most ISPs too have the all-in-one modem/router combo devices. Those, because of the modem, can be harder to switch to because you have to sometimes fight your ISP as to why you'd like to use your own stuff (they'll INSIST that you use/buy/rent their stuff from them). As a matter of principal, I just don't like being told what I can use and what I can't- especially when I'm sure the upgrade I want would fix or add features I'd want. We as hardware enthusiast though don't make up most the ISP's customers, so that's another reason it's harder to get what you want. Customer service don't get why you'd want to use your own stuff basically, so there's a lot of "WHY U WANT USE DIFFERENT, YOU HAVING ISSUE?".
Yes, the hardware/software choices, I get that. One of the reasons would be just that. I noticed that most routers on sale are not modem/routers so you would have to buy a modem and a router and... what? Let's say you disconnect the coaxial cable from your ISP's modem/router and connect it to a modem and then connect that to a router, won't it just work out of the box, what kind of issues could you have?
Well, for one, obviously this is all different depending on dsl/cable/sat and then what ISP you have. I'm assuming you're talking basic cable internet, right? The router part is easy, it'll just work. The modem though, it could take up to 5 days for your ISP to "auto" connect to your new one. That was my experience, even after calling tech support and telling them I had just installed a new modem. They just told me to wait, and wait I did, for around 4-5 days till it just one day connected. After it did I never had any issue though. I don't regret for one second getting my own modem. Was DOCSIS 3 compatible before they even offered it in my area, and once they did, I was automatically switched to it while everyone else had to get new modems.
Reading How to choose the best router for your connected home should give you an idea on some differences amongst various routers.
I've seen reports of ISP provided [combo] routers not supporting "customer induced public IP Address/prefix changes via MAC Address changes". IOW, users of said ISP provided gear reported less... and in some cases ZERO... control over their public IP Address/prefix assignment. The potential consequences of that would depend on other factors, but the bottom line is that it is an important consideration when selecting gear (and ISPs).