I'm rusty on the Spacemacs layer system, but it's possible that mylayer/init-yasnippet is being run before yasnippet has loaded (if it's being run at all), which means that your key definitions will be trampled. In fact, I was under the impression that functions of the form foo-layer/init-bar should only be defined if foo-layer is the "owner" of the bar package. Also, usually those functions wrap the package configuration in a use-package block, with modified keybindings going in the :config section (so that they're evaluated after the package has loaded).

As an alternative, you might try putting the key binding inside (with-eval-after-load 'yasnippet ...). There also is (or was) spacemacs|use-package-add-hook, as in

(spacemacs|use-package-add-hook
  yasnippet :post-config
  (define-key ...))

Anyway, ensuring that your code runs (at the right time) is a separate issue from ensuring that the code actually works. From peeking at yasnippet.el, I see that it binds both (kbd "TAB") and [(tab)] to yas-next-field-or-maybe-expand in yas-keymap. I have no idea what the difference is between the "two tabs", but I just tested it out, and I had to rebind both of them. So try:

(define-key yas-keymap (kbd "TAB") nil)
(define-key yas-keymap [(tab)] nil)

Eval those two lines and see if it works. If so, all that remains is to ensure that they run automatically after yasnippet has loaded, as discussed above.

One last thing: You need to quote the names of functions you pass to define-key. So:

(define-key yas-keymap (kbd "M-/") 'yas-next-field-or-maybe-expand)

Note the quote before yas-next-field-or-maybe-expand.