Yes, the EU will suffer as well, but not as much as the UK.
Incidentally, Pascal Lamy, the former WTO chief, famously said that leaving the EU was going to be like taking an egg out of an omelette. He also said, more recently, that if the UK left on WTO terms, it would be like going from 1st division to 4th.
https://twitter.com/rjbarfield1/status/1071423059247161344
We have been paying around £9 billion net per year membership fees to the EU.
Even if we help businesses out with around 5 billion Euros in tariffs we would still be approx. £4.5 billion better off each year. As that is roughly what we get from the EU in grants, we could use that to replace them,
EU countries would have to increase their membership fees to cover the loss of our contribution and they would have to pay tariffs too if they wanted to continue to sell to the UK. Germany, as the largest contributor too the EU budget and a large exporter to the UK would be particularly affected.
It's by no means certain that the UK would suffer more in my opinion.