When we bought the first flute for our daughter, her teacher recommended open keys and the difference was even hearable for us as uninformed listeners, even if their main purpose is microtonal adjustments in pitch by partial covering.
There are two further reasons, which I consider as striking: You are enforced, to set your fingers carefully and correct from the beginning (otherwise the hole will not be closed properly and intonation will clearly indicate this).
Closed keys allow a more sloppy approach.
There are sets of small plugs available (and sometimes even supplied with the flute), which convert open keys into closed ones.
(According to my knowledge, their use is only reasonable for children.)
The other direction is much more difficult and means exchanging the keys.
music.stackexchange.com/users/2600/guidot
Having only experience with open holes, I can't comment on closed.
But it seems that the latter are geared towards child beginners.
I have pretty small hands (well, not that pretty...) and covering holes has never been a problem.
As a fully grown adult, you might as well start as you mean to go on - get an open hole one, and just take care.
You'll soon get the message if you're not closing properly.
If you're like me, you'll have more trouble with the embouchure than fingering!
Good luck.
It's one of the more difficult instruments to learn, I think.
Carl's price quote seems quite steep.
There are lots of pre-owned ones up for grabs, and not a lot to go wrong, unless they've been dropped - but that's the same for most musical instruments.
music.stackexchange.com/users/2808/Tim
A quick look at a couple retail online shops indicates you'll pay (for Yamaha, a good line of starter flutes) around $900 -$1000 for a closed-hole, and maybe $500 more for an open-hole flute.
If the money doesn't bother you, then go with open-hole for the reasons given in other answers and comments.
If you're not sure you are going to play the flute "forever," then you may want to go cheap.
music.stackexchange.com/users/6275/Carl Witthoft
I would go with the closed hole flute for two particular reasons.
First of all, it is cheaper than the open hole flute.
This is easier for you if you decide that you would not like to keep the instrument.
Secondly, open hole flutes require a more precise hand position than the closed hole, making it more difficult fingering wise no matter the hand size.
However, the open holed flutes do allow a much more mature tone.
And if you feel your fingering is not precise, there are plugs you can buy to close up the holes until your positioning has improved.
If you are dedicated to playing the flute then the open hole is a better choice.
Most beginners will get a closed hole because they are not sure they will continue playing the instrument.
The basic answer for you can be determined by
which of these qualities is more important to you.
I hope you enjoy playing the flute as much as I do!
music.stackexchange.com/users/49422/Abi1507
With a French Horn, a beginners starts on a Single horn and progresses to a Double horn.
The single is cheaper and simpler.
A general rule for beginners - especially children - who may very well decide the flute is not for them and take up some other instrument, is to start cheap and simple.
As Dr Klee says "most professionals use an open hole Flute".
There is your answer.
You start simple and progress to the more complex, more expensive instrument with the bigger range and tone as the learner becomes more proficient and can attempt more complex pieces.
music.stackexchange.com/users/54719/Landers
A few more things I want to add are: With open holes, a flutist can produce a wider range of special effects, which is common in modern music.
Like, he can cover only part of the hole and alter the pitch this way.
Closed holes flutes have a bit brighter sound and you can move your fingers around a bit without losing the note, whereas in the closed holes the tone is darker.
music.stackexchange.com/users/7356/Shevliaskovic
By looking to various flutes online, I saw that there are two types of flutes:   Open Holes Flutes  Closed Holes Flutes   As an adult beginner, what should be taken into consideration before deciding  which category of flute to buy from?  I can understand that for children, it might be easier to buy a flute with closed holes, because their fingers might not be big enough to cover the holes, but this is not the case here.
Choosing a flute for an adult beginner: Open or closed holes?
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<beginner><flute>
music.stackexchange.com/questions/44361
music.stackexchange.com/users/7356/Shevliaskovic
Open hole flutes are considered better for beginners.  You are enforced, to set your fingers carefully and correct from the beginning (otherwise the hole will not be closed properly and intonation will clearly indicate this).  Closed keys allow a more sloppy approach. Moreover, with open holes, a flutist can produce a wider range of special effects, which is common in modern music. Closed holes flutes have a bit brighter sound and you can move your fingers around a bit without losing the note, whereas in the closed holes the tone is darker. There are differentiations when selecting a flute with open or closed keys
There are two further reasons, which I consider as striking:  You are enforced to set your fingers carefully and correct from the beginning. Closed keys allow a more sloppy approach. There are sets of small plugs available (and sometimes even supplied with the flute), which convert open keys into closed ones. With open holes, a flutist can produce a wider range of special effects. The other direction is much more difficult and means exchanging the keys. Closed holes flutes have a bit brighter sound and you can move your fingers around a bit without losing the note.