What is the difference between a min-heap and a max-heap?

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The only difference is which extreme sits at the root. A min-heap keeps its smallest value on top, and a max-heap keeps its largest. Both obey the same shape rule and the same parent-child ordering, just flipped.

In a min-heap, every parent is smaller than or equal to its children. In a max-heap, every parent is larger than or equal to its children. Neither orders siblings, so you cannot read a sorted list off the array.

You pick based on what you pull first. To take the cheapest task next, use a min-heap. To take the highest-scoring item, use a max-heap. Peeking at the top costs O(1), and removing it costs O(log n) either way.

If your library only ships one kind, negating numeric keys fakes the other.

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Likely follow-ups

In a min-heap, how fast can you find the maximum, and why? The max must sit in a leaf. About half the nodes are leaves and they have no order among them, so finding it is an O(n) scan.
Your language only ships a min-heap. How do you get max-heap behavior? Negate numeric keys on push and negate back on pop, or pass a reversed comparator. Negating the minimum value of a fixed-width int overflows.
To keep the K largest items from a stream, which heap do you use? A min-heap capped at size K. Its root is the smallest of the top K, so you evict the root when something bigger arrives. O(n log K) total.
How would you track the running median of a stream with heaps? Use a max-heap for the lower half and a min-heap for the upper half. Keep their sizes within one of each other and read the median from the roots.
Heapsort sorts ascending in place. Which heap does it build, and why? A max-heap. Each swap moves the root to the end of the array, so the sorted tail grows as the heap shrinks and no extra memory is needed.

What you can say

The only difference is which extreme sits at the root. A min-heap keeps the smallest value on top, and a max-heap keeps the largest.
Both have the same shape and the same parent-child rule, just flipped. In a min-heap every parent is smaller than or equal to its children, and in a max-heap it is larger or equal.
Neither one orders siblings, so the array is not sorted. The top is the only position you can be sure about.
I pick based on what I need to pull first. If I want the cheapest task next, I use a min-heap. If I want the highest score, I use a max-heap.
The costs are the same either way. Peeking at the top is O(1), and removing it is O(log n).
A heap only answers for one end, so if the library ships just one kind, I negate numeric keys to fake the other. That trick works for numbers, not for strings.

Weak answers to avoid

Says a min-heap is sorted in ascending order Only parents and children are ordered, never siblings, so the array is not sorted. Say the root is the one position whose value you know for sure.
Describes a heap like a BST, smaller on the left A heap has no left-right rule. The order runs from parent to child only, so reading the top is O(1) but finding any other value takes a scan.
Treats them as two different structures Same shape, same sift-up and sift-down, same costs. Only the comparison flips. Peek is O(1) and removing the top is O(log n) in both.
Defines both but never says when to use which Tie the choice to what you pull first. Taking the cheapest task next calls for a min-heap, and taking the highest score next calls for a max-heap.

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