Many apps, including Illustrator & Photoshop, will automatically replace straight quotes "" by curly quotes “” or the straight apostrophe ' by the curly one ’ or even other quotation marks depending on your language preferences.
So if your font contains only the straight quotes characters, they will probably be displayed with a different default font. There are a couple ways to solve this:
Create additional glyphs for the curly quotes.
Or disable the automatic replacement feature in the app where fonts are used.
1. Create additional quotation marks
First, identify the additional marks you need (check this Wikipedia article to find which ones are required for different languages).
These are the most frequent marks:' " ‘ ’ “ ” ‛ ‚ „ « » ‹ › `
Once you've made the choice of which quotation marks you want to support in your font, select and copy the text of each mark, and paste it under each specific glyph, like below:
PRO TIP (time-saver when creating many glyphs)
To speed up the process of creating many quotes at once, use the Smart Batch feature by following this tutorial (you can either name each object in Illustrator's layers panel and press Batch or enter comma-separated values).
2. Disable the automatic substitution
In Adobe apps, uncheck Type > Smart Punctuation... > Smart Quotes
In Pages or Keynote, uncheck Edit > Substitutions > Smart Quotations
In MS Word, uncheck File > Options > Word Options > Proofing > AutoCorrect > AutoFormat > "Straight quotes" with “smart quotes”
Or on newer versions, uncheck Word > Preferences > AutoCorrect > AutoFormat as You Type > "Straight quotation marks" with “smart quotation marks”