No, you don't need and I don't think you can install both anyway, they usually get in conflict. You need to choose one.
Which one you choose depends on what you need and prefer. OpenTypes are like an "improved" version of TrueTypes.
If you're using an old Windows system, the OpenType might not work unless you tweak the registry. If you do web design and want to have access to expanded characters, OpenTypes offer this and they're lighter than TrueTypes. The hinting is not processed the same way and apparently TrueTypes offer nicer results for very complex fonts. As as you'll read on that link, some legacy software or system don't support OpenType hinting.
And there's the expanded character set (swash, ligatures, fractions, unicode, different alphabets, etc.) of OpenTypes that can be easily accessible in software like the ones offered by Adobe. You might prefer to use a quality OpenType if you need to have access to complex mathematical symbols or need to work on projects related to linguistic for example (Related: What are those things in text design?). But even if you get an OpenType, it's not a guarantee you'll find these extra characters if the font wasn't created with them from the start.
For printing quality, they're both good.
One thing though, even if OpenTypes can contain more characters than
TrueTypes, it doesn't mean they actually do. Your 2 sets could be identical. You'll need to verify if your OpenType has the extended sets by trying it!