In most cases it's always better to leave. Don't let comfort zone and fear stop you from expanding your horizon.

The few occasions where you don't leave could be:
1) Current company is top 3 in the industry and plenty of room to grow.
2) Offer from new company is too low, or logistics (ie- need to move faraway etc) are not great.
3) Current company offers a promotion and expanded scope.

I'd advise going, and don't bother with any counteroffer. After all, you're familiar with everything - meant things are getting stagnant. Learning new things, meeting new people etc outweights the risk of joining a shit company or a bad boss. If you're really unlucky, you could seek another opportunity elsewhere.

Folks that do well in career are constantly learning and upgrading by doing something or going somewhere different.

Also 20% is too low... but it does depend on the leverage you have. All the best deciding!

Because HR arent there to pay you fairly - they're there to ensure they pay you the minimum possible. That's why they needed the written offer to set a baseline and increase minimally or match. If your boss or HR values you, they'd offer what you're worth, not match what other companies think you're worth.

In one of my last company - I told my boss I am underpaid and on the spot they increased my pay. No third party offer or letters needed. They understand if they don't, I'll leave. BTW I left anyway....