When I give a button and a link the same styles, why is the link one pixel higher?

It's not logical, so don't be too hard on yourself. You need to give it a height and change the box-sizing and display:

button
input[type="reset"],
input[type="button"],
input[type="submit"],
input[type="file"] > input[type="button"]
  display: inline-block
  box-sizing: border-box
  height: 20px
  line-height: 20px

Also see Proper cross-browser CSS styling for buttons.