Exodus vs Core Wallet vs Kraken

So this is kind of two part question

What (if any) is the difference between using a desktop or mobile wallet like Exodus versus using the core wallet?

Second question is what makes a desktop wallet more secure (or is it even more secure) than keeping your crypto on kraken with maximum security. Maximum security means you have 2 factor and a master password enabled.

This kind of strikes me like keeping your cash at home versus in the bank. I would feel safer with my cash at the bank that spends tons of money on security versus an apartment behind the security gate and a locked front door (even if you take away the FDIC nature of a bank I think it's safer than your home).

Happy to be wrong and have someone teach me!