Buying someone a big expensive toy is always fun. But this isn't a toy - it's a tool. People use cameras for so many different things that this tool has been customized into hundreds of configurations with different pros and cons at all different price points - so you can find your perfect one.
You should not buy your friend this gift as a surprise.
Instead, you should tell her that you would like to support her photography but only if she wants to learn more and grow as a photographer.
She may be perfectly happy with the smartphone. Or, you'll end up having a conversation about what is important to her.
Now that you know what's important to her, you can go shopping for product advice.
Don't make this a surprise gift, instead, be a part of the journey - if you really want to help. You may find that your initial assumption about a compact camera is dead wrong.
I gotta say, I'm getting a lot of feedback regarding people's opinion on whether I should get this gift at all, but that's completely besides the point of the question. The question posed is regarding the value of a smartphone or a camera for a beginner photographer. Lets just take it as a fact that the person wants to get into photography and wants camera advice. (Perhaps a daughter asked her father for help on this issue...) No offense, but your answer has been the most upvoted of all the given ones here, yet it addresses the posed question the least
Your question is not simply, tell me the difference in cameras. You are basing your purchase decision on a lot of assumptions - the biggest one apparent in the way you wrote your question is that your friend wants to grow as a photographer. That is a huge assumption that makes the rest of the question mute; it's also the part of your question that I chose to address. Clearly, I'm not alone in my assessment.
But, alright - let's take your assumption.
No I would say that that's too specific, though that's not to rule it out. capturing moments might be better e.g. family moments. but for example I would imagine that she would use it a lot when traveling.
Family snaps were the hallmark use of P&S disposable film cameras, then P&S digital cameras, and now cell phones. If this is the primary use case - my advice is to stick with the phone. Ease of use and share-ability will trump the (small) gain in resolution.
As for travelling - the same may hold true. How likely is she to want to share photos over take and then edit them (or at least transfer to a computer first)? Is carrying more gear a problem? When you say travel photos, are landscapes likely to be in the use case or are we still talking about street photo and snaps?
As you can see from the above - there are many more questions to drive this purchase decision. And only she will have the answers.
IMO - a multi-purpose device that can capture snaps, is wildly user friendly, and allows near instant upload for share/backup (a phone) beats a compact camera any day of the week.
If your friend is truly looking to grow as a photographer - then a mirrorless ILC or SLR will be the definite way to go. Though, there's potential that a bridge camera could suffice for a while.
Spending less than $600 to dive into this world is very possible, especially if you hit the used market. I got lucky with a 5dmkII for $600 in pristine condition - though you're more likely to find 40/50/60D's below that. Toss in a 24mm pancake lens, a 50mm, and a 85mm - or a kit zoom and you've got a base kit that'll take you far.
The absolute best thing you can do is come up with a list of the top 5 things she wants to shoot, and get her opinion on the pros and cons of a compact, mirrorless, slr, and bridge as it pertains to her future use cases.
Then you can make an intelligent purchase decision. I'm sorry if this still doesn't help you. It's very hard to make a recommendation when there are so many assumptions/speculations about the buyer.