Rattlehead

And with a furious man do not go,

Lest you learn his ways

And set a snare for your soul.

We're all human, and we all make mistakes. Even if both you and the other party contributed to the tension or misunderstanding, when you're the first to apologize it demonstrates maturity and responsibility. Whether it's professional relationships such as with co-workers, clients, or suppliers, or personal relationships such as friends, romantic partners, or family, people will generally prefer to keep and maintain relationships with someone capable of demonstrating personal accountability.

It also defuses the tension sooner when you just get it over with and apologize first, rather than letting the tension drag on and on while you both wait for the other person to say something first. Notably, if you apologize first and diffuse the tension sooner than later, the diffusion of tension can lower the other person's guard, allowing them to reflect on their own words and actions, and hopefully own up and apologize to their own part in the tension / misunderstanding.

This doesn't mean you need to take all of the blame unjustly, so keep your apology limited to what you may have contributed, but you don't have to extend it to the whole situation. Examples of apologies that keep it just to your own contribution are "I realize my tone might have come across as too harsh or sharp earlier, and I'm sorry for that. I didn't mean to come across that way."