Photoshop font looks diffrent then Affinity Photo

Just to clarify: Why do you want to use inch-marks instead of the »typographically correct« quote marks?  

@PBArtattack wrote:

Edit > Preferences > Type... > turn off "Use Smart Quotes".

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That is an OK solution, but professional typographers would have a problem with making Photoshop always use inch marks instead of actual quote marks.

If you prefer to override the quote marks on a case by case basis, it would be better to use the Glyphs panel (View > Glyphs), where at any time you can choose from all character variations available in that font. Double-click the glyph you want to enter it in the text. Below, the Glyphs panel is set to show Punctuation only, and it has inch marks, quote marks, brackets used for quotes in other languges, etc.

As Jeff Arola mentioned, the exact appearance of the type edges can be affected by which Anti-Alias option is selected in the Character panel. See if any of the choices match Affinity Photo better, or see if Affinity Photo has a similar option.

The difference in apparent size could be due to different anti-alishing between the two programs.

Edit > Preferences > Type... > turn off "Use Smart Quotes".