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LingakuLingaku[subject] + 以为 + [clause]
[subject] + 认为 + [clause]
以为 and 认为 both take a clause, but they say different things about it. 以为 reports what someone believed and implies it was wrong. 认为 states an opinion, usually a considered one, and says nothing about whether it was later disproved.
With the same clause, the verb alone decides whether the speaker still believes it.
还以为…呢 in speech adds surprise at being wrong.
To ask someone's view, use 认为 or 觉得. 你以为…？ sounds like a challenge, "do you really imagine…?".
In formal writing 以为 can still mean "consider, hold the view", as in 本人以为 "in my view". This is the older sense and appears in essays and official statements.
Stating your current opinion with 以为 tells the listener you've changed your mind.
The negative is 不认为. 不以为 survives only in the idiom 不以为然 "disagree".
觉得 is the everyday spoken choice for "I think"; 认为 is the firmer, more formal one.