The normal expression to use is "by heart". However, both sentences are valid and they mean the same thing. Neither sounds excellent. To make #1 work better, put the adverbial phrase "from memory" closer to the verb ("knew") that it modifies. The problem with putting it at the end of the sentence is that it appears there to modify "the songs": "From memory they knew the songs." (best) "They knew from memory the songs." (acceptable) Same for #2. If you use "by heart", you can put it anywhere in the sentence: "They knew the songs by heart". The reason "by heart" works even at the end of the sentence is that the expression "by heart" is strongly associated with memorization. Nobody will have any trouble recognizing that it is adverbial and modifies "knew" to express HOW they knew.

They memorised the songs. They knew the songs by heart.

‘From memory’ is good but your example isn’t the best. By definition ‘knowing’ uses ‘memory’ so it’s redundant. Natural are: The pianist can play a thousand songs from memory. (She doesn’t need to look at sheet music) From memory, it cost me about $4000. (I have some doubt. To find the correct and exact amount I would have to refer to my records. ) Replacing ‘from’ with ‘by’ might be understood but is much weaker.