It happens a lot. You like traditional style while your boyfriend/girlfriend/roommate loves modern design. You end up bickering over furniture choices and purchases. A new rug becomes the launching pad for a week long fight. Maybe it's an extreme example, but it does happen, and there are ways to get around it and keep both parties happy with their space.
Get Your Henry Clay On
Due to his impressive negotiating skills, Henry Clay was a Kentucky senator who was dubbed The Great Compromiser. He helped keep members from both parties happy while still actually accomplishing things. This is exactly who you'll want to channel when it comes to making design choices when there are conflicting style preferences. Know that having a strict style, whether traditional, modern, or mid-century modern just isn't as realistic as some magazines or HGTV shows make it seem. It's important to keep styles livable and to be flexible, thus making strict styles more difficult to incorporate. Just being open to compromise is step one, and it has to happen unless you want to continue to bicker about it.
Find the Venn Diagram Happy Place
You know where the circles overlap in a Venn Diagram? The place that lists things that each circle have in common? That's what I like to call the 'happy place of a Venn Diagram" and it's an area that you want to focus on first. Even styles that appear totally opposite will have some overlapping style characteristics and colors that you both might just love. Focus on incorporating those elements right away. These are things you both agree on and things that you both love. A good example might be button tufting on chairs – while technically more traditional, it's also often seen in modern designs as well. You can find seating that incorporate button-tufting in a neutral color to appease both preferences.
This can also help keep the style cohesive, even if not necessarily fitting in with the strict style definitions that exist today.
Consider What You Really Care About
Taking whatever each of you are passionate about and somehow incorporating it into your home design is really important. This helps to make the space uniquely yours and will make each party feel more comfortable and more at home. If one of you loves to cook, maybe allow that person to have free reign over the kitchen. For my own personal example, I love books. They are present in every nook and cranny of our Kensington place. Andy, meanwhile, loves music. His instruments, including his guitar and his trombone, are all on display throughout the apartment as well. Sometimes, displaying these passions makes us care less about the other stylistic choices, as something that is very "you" is already present.