Why padding is included in height sometimes?

That depends on what box-sizing attribute you have used:

border-box means that the height and width of the box, defined/calculated in CSS, will also include the padding(s) and border width(s) applied to it
content-box is the default behavior, where padding(s) and border width(s) are added onto the defined/calculated height and width of the box.

By setting box-sizing: border-box as seen in your left example, you have defined the height of the element at 20px. This means that the actual content box will only be 8px tall, because the browser will subtract the border (1px top, 1px bottom) and padding (5px top, 5px bottom) form the defined height, leaving only 8px left, which is a tad bit too short to contain height of the entire line (therefore the word appears to be cut off).

What is the difference between border-box and content-box in CSS?

While box-sizing: border-box; uses the box-model that people have come to associate with Internet Explorer, where the dimensions of the padding and border are included in the element’s dimensions.

https://jsfiddle.net/ud25tGNwMddCP7JI/25gtsr1d/17/ 参考这个demo

box-sizing

The box-sizing CSS property sets how the total width and height of an element is calculated.

By default in the CSS box model, the width and height you assign to an element is applied only to the element's content box. If the element has any border or padding, this is then added to the width and height to arrive at the size of the box that's rendered on the screen. This means that when you set width and height, you have to adjust the value you give to allow for any border or padding that may be added. For example, if you have four boxes with 25%;, if any has left or right padding or a left or right border, they will not by default fit on one line within the constraints of the parent container.

The box-sizing property can be used to adjust this behavior:

content-box gives you the default CSS box-sizing behavior. If you set an element's width to 100 pixels, then the element's content box will be 100 pixels wide, and the width of any border or padding will be added to the final rendered width, making the element wider than 100px.
border-box tells the browser to account for any border and padding in the values you specify for an element's width and height. If you set an element's width to 100 pixels, that 100 pixels will include any border or padding you added, and the content box will shrink to absorb that extra width. This typically makes it much easier to size elements.

Note: It is often useful to set box-sizing to border-box to layout elements. This makes dealing with the sizes of elements much easier, and generally eliminates a number of pitfalls you can stumble on while laying out your content. On the other hand, when using position: relative or position: absolute, use of box-sizing: content-box allows the positioning values to be relative to the content, and independent of changes to border and padding sizes, which is sometimes desirable.

Syntax

The box-sizing property is specified as a single keyword chosen from the list of values below.

Values

content-box: This is the initial and default value as specified by the CSS standard. The width and height properties include the content, but does not include the padding, border, or margin. For example, .box { 350px; border: 10px solid black;} renders a box that is 370px wide.; Here, the dimensions of the element are calculated as: width = width of the content, and height = height of the content. (Borders and padding are not included in the calculation.)
border-box: The width and height properties include the content, padding, and border, but do not include the margin. Note that padding and border will be inside of the box. For example, .box { 350px; border: 10px solid black;} renders a box that is 350px wide, with the area for content being 330px wide. The content box can't be negative and is floored to 0, making it impossible to use border-box to make the element disappear.; Here the dimensions of the element are calculated as: width = border + padding + width of the content, and height = border + padding + height of the content.