Safari, the stock iPad browser, is essential and well done but intentionally crippled in two ways: no Flash and no add-ins.
Although some say Flash is dead, it lives on. Surprisingly enough even parts of Amazon.com still have Flash. It is in places like “Manage my Kindle”. Also, I use Training Peaks for my training log and it is Flash based. So when I need I flash, I use Puffin. I have yet to see a site that it couldn’t handle.
Most browser add-ins are just convenient, but LastPass is mission critical. If you are going to be even somewhat secure on the Internet you need to use a different random password for every site you visit. My choice for a password manager is LastPass. It rocks! On everything but iOS I use a browser add-in. On iOS thankfully, the LastPass folks have created LastPass tab. It is an adequate browser that fills in all my logins and passwords via my LastPass vault.
So how does this all work out? Most of the time I use LastPass tab. When a need Flash, I switch to Puffin. And when I need some integration with the Apple ecosystem, I use Safari.