Being constitutional monarchs they have no day to day power, but they can refuse to assent to any proposed new law, thereby preventing the political party in power from getting carried away, etc.
Also, there is no point in assassinating a monarch as the next in line automatically succeeds to the throne. (Unlike assassinating a president.) Then of course, there is the huge cost and corruption involved in standing for the presidency, with elections every 4 years. Plus the cost of lifetime security for all past presidents and their pensions, allowances et al.
The monarchy is also well trained in handling visiting dignitaries and heads of state, unlike certain US presidents.
All in all, it is a remarkably cost effective and a superb tourism draw.
Over and above this, most British people simply like having a benevolent monarch.

We have actually evolved from Kings and Queens who had horrible violence and over obeisance pass history. Today Kings and Queen create an hierarchy delusional of fantasy of freedom and rights. Just far less in action and power as the political leaders powers of today. I call them both fairytale of an ugly freak show that started out in a war over 3% tax on tea based from the British kings to America. Today we can't stop the bleeding of 35% to half your money in taxes, liences, lies and more than 75% of their broken promises.

To me, there is no draw, no appeal, no significance except that which delusional types hold dear to their hearts. She was a meaningless figurehead and, as far as I can tell, had very little impact or influence outside her immediate orbit. I have nothing against her, but the monarchy is an irritating anachronism which was replaced with a far better political system (though far from perfect).
If it were up to me, the British monarchy would be abolished. But I don't really care one way or the other.

I've been thinking about this, too...what do Americans think about the
Queen?

This is a personal perspective, of course.

She was a young woman who found herself in a position of maintaining a centuries' history....and she did it.

She was a young woman who spent her entire life in service to her nation. She promised that in an announcement..."I will spend my entire life...long or short....in your service."

She was a middle-aged woman who recognized change in her culture was imminent.

She balanced her position with leaders of other nations....and they always seemed to respect her.

She had to stand up when her children broke with traditional guidelines of British culture.

She got through it all.

I've heard she was a talented mimic....imagine her mimicking the foreign leaders she'd met?

She loved dogs and horses.

I hope she had happiness in her life...she kept her bargain with the people of the UK.