Do 'learn by heart' & 'learn by rote' mean the same?

No. They are not the same. In fact, they are not even related much. Sadly though, the expressions came to be used interchangeably and even Wikipedia has merged them into a single entry.

Learning by rote is about the technique or practice of memorizing. Learning by heart on the other hand, is the nature or quality of what has been learned.

The director told me to learn my speech by heart. I had to go over it many times before I learned it by heart. (TFD)

This clearly shows the difference. go over many times is the process of 'rote learning'; I learned it by heart is the effect it produced or the quality of learning that was acquired.

learn something by heart
Fig. to learn something so well that it can be written or recited without thinking; to memorize something.

Compare:

learn something by rote
Fig. to learn something by memorizing without giving any thought to what is being learned. I learned history by rote; then I couldn't pass the test that required me to think. If you learn things by rote, you'll never understand them.

Do you notice the without thinking part? It occurs in learning in case of 'rote learning', and in recollection in the case of 'by heart'!

When I was at primary school in the early 1950s we were encouraged to learn things 'by heart'. For example we learned all our times tables from 2 to 12 'by heart'. We learned poems by 'heart'. At that time the practice had not come under the sustained criticism of educationalists that it has today. And the expression 'learning by heart' had a nice gentle ring about it.

I never encountered the term 'learning by rote', nor 'rote learning' until the practice started to become derided by people who claim to understand these things almost as though it were harmful to children's health.

My own suspicion is that whilst they mean the same thing, learning 'by heart' sounded too wholesome and positive a learning experience for those people who wanted to get rid of the practice. So an altogether less enticing term was introduced. I may be completely wrong about this, but that is my impression.

Is it wrong to conflate the two? I think that would largely depend on who you were talking to. Personally I have never understood why 'memorisation' excluded the possibility of 'understanding'. After all actors and actresses learn their parts 'by heart', or is it 'by rote'?

I would say the terms (which mean the same thing) are used by people depending on what impression they wish to create. Whilst Shakespearean actors probably memorise 'by heart' (because that is deemed nice), children are taught times tables 'by rote' (because that is deemed nasty).

Further thought.

@Damkerng T (below) has suggested that learning 'by rote' is only one method of learning 'by heart'. If that idea has substance it suggests that 'by heart' is a more general term and that 'by rote'is a subset of it. Could it be that 'by heart' refers to an end-result e.g. I know the 23rd psalm 'by heart'. And could 'by rote' refer to just one process by which you get there?