how the text looks in the doc
how it looks in html
how the text frame looks in the doc vs html
If you're exporting with Text Rendering set to one of the HTML text options, check to make sure that you don't have any style overrides. You can view them by going to the Character Styles panel menu or the Paragraph Styles panel menu and selecting Toggle Style Override Highlighter. When that option is enabled, text that is formatted with overrides to the styles appear highlighted in cyan.
If you export with in5 and Text Rendering is set to Images, the style overrides aren't an issue.
However, when you export using HTML text, in5 uses the Character, Paragraph, and Object Styles to create the CSS. If there are overrides, then in5 goes through a series of checks to try to "guess" what the style should be. These overrides can cause unintended results. So if you'd like to export with HTML text and the HTML text styles aren't rendering as expected, clear any style overrides.
Here's more information on Text Rendering: