Why do I react in one way to a stressful situation and my partner reacts in a completely different way?

It is interesting how 2 people in the same situation can react so differently!  I love seeing how differently silver medallists in the Olympics react, some totally devastated, and others ecstatic.  Why is this so? Well, the main (broad) reasons are

1. We interpret whatever is happening - if a silver medal is interpreted as “I've let my country down”, or if it's interpreted as “I'm the 2nd best in the whole world”, that will trigger quite different reactions.  The way we interpret things that happen is largely based on assumptions we hold about ourselves, other people, the world, and the rules that we have about life.  These assumptions are learned.

2.  We have learned patterns of responding.  Two people might interpret a situation in exactly the same way, but they might have learned to respond in quite different ways.  For example, if you're having trouble with your mobile phone company, one person might respond in a calm way, thinking that that's the best way to handle it, while the other person believes that you need to shout at people so they know you're serious.

There are other factors that come into play, such as hormones, fatigue, general health, and whatever else is happening in your life at the time. 

If you're not satisfied with how you &/or your partner are managing stressful situations with your kids, you could try seeing a counsellor / psychologist who can help you to understand what's making the situations so stressful for you, and hopefully to find more effective ways to manage them.  That doesn't mean that there isn't a place for being serious, and conveying to your kids how you're feeling, but if you find that you're too worked up too often, then it might be worth seeing someone.  

The funny thing about “why” questions is the way that, when answered, there’s always another why question underneath. And another underneath that. Joe has given an excellent answer about differences in the way we interpret a situation and learned patterns of responding. If I was going the “why” path, then I’d want to ask “why” do I interpret the situation differently? why have I learnt a different pattern of responding?
Instead of responding to your “why” question, I wonder “how’ and “what”. How do you want to respond when something stressful happens with the kids? What sort of mum do you want to be? What qualities do you want to model in the face of stress? How do the mums you really admire & respect react/respond to stressful situations? The answers to these questions might help you decide if you want to learn a new way of reacting/responding.
You’ll notice I’ve said “react/respond”. They’re different. We can react. Or we can respond. Reacting is like a button being pushed and we’re off! Absolutely necessary when there’s a speeding car bearing down on one of our children. Really useful in the presence of any immediate imminent threat of physical harm.  Responding is a more deliberate intentional behaviour. Useful in the presence of threats and stressors that don’t involve physical harm. Maybe you and your partner have the perfect complementary combination. Or maybe each of you can learn something from the other.
The problem with being “reactive” is that it pumps a lot of stress hormones around the body (esp cortisol and adrenaline). These hormones are good in moderation, or in short bursts, but damaging to internal organs in excess or over longer terms.
If you want to learn to be more responsive and less reactive (or your partner wants to learn to be more reactive and less responsive), I’d encourage you to consult a psychologist. As Joe has said, you can learn new ways of being. Parenting is, in my opinion, just about the hardest thing we do. And probably the most important. What do you have to lose by asking the advice of a professional?
I wish you well on your parenting journey.

Joe has given a great breakdown of how our reactions are a result of our interpretations. 

My response is simpler than that. I have had many couples tell me that they are very different to each other as if there is something wrong with that. In truth, every couple will have differences because you each are unique individuals with different genes, family backgrounds, culture, history, personality type and sometimes gender. 

The important thing when parenting is not to criticise each other for the differences but to value them. If you are hysterical, show appreciation that your partner is calm and maybe even hand over the situation and walk away. 

Learn from each other and do not compete. You both care about the wellbeing of your kids. When you synergise your differences 1+1=3.

Viva la difference!

Some great thoughts here given by others. I wonder, though, what it is you're reacting to? It's great (and important!) to think about being responsive rather than reactive. However, there's something in the way you've phrased your question ("when something stressful happens to our kids...") that makes me wonder if your reactions are about fears for their wellbeing or safety in some way? I'm also curious about what life was like for you as a kid - did 'something stressful happen' to you, or to someone close to you, at a time when parents or other caring adults weren't available to protect or help you/them? Unresolved or unprocessed wounds from the past can show up in the present in the form of a heightened reaction to situations that may resemble the difficulties we've had in the past. If this is the case, it won't be terribly helpful to focus only on getting control of your reactions. What WILL help is to understand why you are reacting as you do, and perhaps to process the feelings and thoughts associated with events from another time in your life, that are impinging on your life in the present. Counselling can help you to make sense of this, and provide a safe space to connect to and work through what may lie beneath your current reactions.

I agree with all of the great responses above. Put simply I would say 1) Value the differences and acknowledge that neither is prefect for every situation. Maybe you can practice 'borrowing' a bit of your husband's calm, and he to get a bit more responsive. We choose our partners for unconscious reasons, some of which include being able to bring more of the qualities we don't have enough of into our family system. This works both ways. 2) The meaning we make of things is due to our own life experience and it may be worth thinking about what you experienced at the same ages that your kids are now.

Counselling is a wonderful way to learn and grow and to become more aware of how to change if that's something that you want to do. All the best.