Upsides are traditional banks usually have insane fees for investment accounts, but with Fidelity (and probably Schwab) you can have it all under one roof. You generally get more options, too. For example, when I opened a 529 I could choose to "open it" in one of several states. My previous big chain bank only allowed me to open an account with the investment options in my state.
There are only two downsides I've seen so far. The first is that Fidelity does not offer personal Health Savings Accounts. This may not be a problem if you have one through your employer, or don't use one. The second is for rare instances when you might need large amounts of cash, larger than you could reasonably withdraw from an ATM. For example, purchasing a car from a private owner. Doing this might require some finagling. In my case, my employer has my 401k at a national chain bank, so I am confident I could cut a check there if I needed to, but it is a concern.
When I had a BofA account, I once asked a teller if I'd need to notify them in advance if I wanted to take out a large amount of cash; when she asked how large she was thankfully able to stifle a laugh and instead say "that should be fine" when I told her "About $5k"
Best to get a local account without any hoops to jump through to keep it fee free in case you have to go to the bank for some odd reason and open up a fidelity or Schwab account for daily use.
Car loans / Home Loans get bought up by Chase and you can service all that sort of thing in one web interface. I'm worried that local banks will make me switch back to snail mail days - I'd have personal access but boy I just want to be able to do it all virtually.
