"Browser security" is about far more than secure connections. Using HTTPS to secure the connection between browser and server is an important part of browser security, but it's certainly not the whole thing. To show what I mean, here are some things off the top of my head (and by the way, I have never done any work related to browsers, these are just things I've heard about):
- Cross-Site Scripting (XSS): If I have two tabs open, can one tab get access to the cookies / session info for the other tab, allowing it to impersonate the other tab through GET / POST requests?
- Media file exploits: Many browsers now include built-in media libraries for things like PDF, audio, video, etc. These are very complex media types with long lists of supported codecs. Libraries that parse them are notoriously buggy for things like buffer overflows that allow arbitrary code execution. How much security analysis have you done on the libraries that your browser compiles against?
- Root Certificates: Most browsers now embed (aka "pin") the root SSL certificates into the source code of the browser. How fast will the browser issue a security patch if one of the root CAs is reported to have a breach?
Those are just examples off the top of my head. Bottom line: "Browser Security" is a HUGE topic.
To answer your other two questions:
- How does the level of security differ from one browser to another?
That depends on which browsers you are comparing. Firefox and Chrome? They are pretty similar. Firefox and the browser my friend wrote for fun last weekend? There's a pretty big difference. Everything else will be somewhere in the middle.
- Insecure connection: browser issue or network issue?
Yes. Also No.
Sometimes it's the browser's fault. Sometimes it's the network's fault. Sometimes it's the server's fault, or the OS's fault, or the user's fault, or the IT department's fault, or
openssl's fault, or the VPN's fault ... the list goes on.
Actually, scratch that, a good security developer will write their code under the assumption that the other machines on the network are malicious (hopefully, including routers). So a faulty network should not be able to cause an insecure connection, but even if it can that doesn't mean it's the browser's fault: it could be any of those other things I mentioned.