Yes, I think "вынести" involves some stronger suffering than "перенести".

We often use "перенести" when talking about some surgery or desease which happened in someone's life. It does not always mean "enduring". If we say "Он перенёс операцию на сердце" it just means that "He has had a surgery on his heart". It just accentuates his experience, but does not emphasize suffering. The same is: "Он перенёс инфаркт" - "He had infarction (and then recovered)". I would add, that "перенести" more emphasizes a lucky result than suffering itself. The person who "перенёс" something has finally recovered from some undesirable conditions.

"Вынести" also says that the person has survived. By it more emphasizes severe suffering, I would say.

But they are often interchangeable:
"Я не вынесу эту боль" = "Я не перенесу эту боль" - "I cannot endure this pain".

But if we say "Он легко переносит боль" it would mean "He easily endures pain". But "Он легко выносит боль" would sound strange to me. "Легко" and "выносит" just do not come together well.