If you look at a Japanese-English dictionary, it's pretty easy to see why these words are often mixed up:
覚える (おぼえる / oboeru)
to memorize; to commit to memory; to learn by heart; to bear in mind; to remember
思い出す (おもいだす / omoidasu)
to recall; to remember
Let's look at the following situations in English, and you can see if you can guess which word we would use, 覚える or 思い出す:
1.
A：彼の名前は何だっけ？
B：あ！タケシだ！今＿＿＿＿＿＿＿！
A: kare no namae wa nan dakke?
B: ah! Takeshi da! ima ______________!
A: What's his name again?
B: Ah! It's Takeshi! I just remembered.
2.
A：彼の名前は何だっけ？
B：さあ...全然＿＿＿＿＿＿＿ない。
A: kare no namae wa nan dakke?
B: saa... zenzen _________________nai.
A: What's his name again?
B: Who knows. I can't remember at all.
The answer's would be:
1. 思い出した (おもいだした / omoidashita)
2. 覚えて(ない) （おぼえてない / oboete nai )
The first one is 思い出した, because you were just able to pull out (出す / dasu) the information that was previously entered （覚えた / oboeta） the first time you learned Takeshi's name.
How to Remember the Difference
I think this is a fairly easy difference to distinguish. First, let's look at 覚える.
覚える (oboeru)
覚える can mean "to memorize," because to 覚える is to put stuff into your brain, put simply.
単語を覚えた。
tango wo oboeta.
I memorized the [vocabulary] word.
In a very similar way, it can also mean to remember. If you think about memorizing, it just means to store information in your brain, which is kind of the same thing as "remembering," no?
The tricky thing here is that it is possible to say 覚えている for a thing that you still remember.
For example, taking the conversation from above.
A：彼の名前は何だっけ？
B：俺は覚えてるよ。
A：じゃあ教えてよ。
B：教えない。
A: kare no namae wa nan dakke?
B: ore wa oboeteru yo
A: ja oshiete yo
B: oshienai
A: What's his name again?
B: I remember it.
A: Then tell me what it is!
B: Nope.
In this example, Person B can say 覚えてるよ, because he has not forgotten Takeshi's name. It is well organized into his brain, much as though he memorized it. This is different from the Person B we saw earlier, who did not memorize it properly, so he had to work to pull it out (思い出す)of his brain.
思い出す / (omoidasu)
This trick to remembering how to use 思い出す is that you can say it whenever it would be okay to say "to recall" in English. By doing this, you can also say that you "don't recall."
For example, we could put a spin on #2 from above...
A：彼の名前は何だっけ？
B：何だっけ？！思い出せない！
A: kare no namae wa nan dakke?
B: nan dakke?! omoidasenai!
A: What's his name again?
B: Ah, what was it?! I can't remember! [=I can't recall.]
Remembering... recalling... memorizing... sounds like trying to learn a language.
Learn more subtle nuances at NativShark.
Good luck with your studies,
Niko
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