I think that there are a lot of different things going on here that need to be addressed in different ways. First, helping the child work out these issues, and second focusing on your relationship with his mother. (Sorry, this is going to be long.)
The first thing I'd say is to remember that your son is two years old. Remember when you're having these interactions that he's not trying to misbehave; he's trying to learn how to navigate his environment, learn how to relate to people socially, and reacting to stimuli. He's not any happier about these tantrums than you are. No two year old is going to be perfectly behaved in nearly any situation.
He's also a second child. Don't compare him to your first - he's in a very different situation than she was. He's only 18 months younger, so she still needs a lot of attention herself; he gets probably 60% of the attention she got at this age. So it's not at all surprising that he's doing a lot to get your attention, and frustrated when he doesn't get it. You can't totally fix this - you don't have 200% of the attention you had before he was born, after all - but you need to take it into account.
Keep in mind when these things happen that your goal when resolving the tantrum is not to remove the annoying noise/spectacle/embarrassment caused by the tantrum, in most situations. If you're in situations that are going to cause excessive embarrassment (at a theater, say), don't do that until you've worked through this period.
When they do happen, and there's both a clear problem and sufficient coherence from your son, focus on solving the problem. If he's screaming because he wants something, remember that at two, the ability to think about the future is very limited. He thinks that he's never going to get a sweet because you won't give him one right now. So help him learn to delay gratification. "Sorry Billy, you can't have a sweet right now because it's close to dinner. But I have a sweet for after dinner that's in my bag here." Then when after dinner happens, make a point to connect it to the earlier experience. This won't work instantly, but over time he'll begin to learn that "no right now" doesn't mean "no forever".
When they happen and there is not a clear problem or not sufficient coherence to work on it, focus on destressing the situation. Remove the focus on the trigger (if there is one). Focus on calming activities. Remove everyone to a quieter area, let the child express his emotions in a place you won't feel as uncomfortable, and be positive and calming. Remember, in these kinds of episodes the biggest issue is that the child cannot express himself effectively, so he resorts to crying and screaming.
Second is your relationship with his mother. You don't specify if you're married, but you say 'together as a family' so I'll treat it as if you are for purposes of this answer; it's not really important, so long as you're co-parenting, anyway.
Things like this need to be addressed through mutual discussion, understanding, and an agreement to proceed in a particular way. You and she need to talk together about how to deal with these situations, and need to come up with a way that works for both of you. This needs to come from both of you - not you telling her how to handle things; but you and she agreeing how to handle things. Sit down and have that discussion now, and at a time that isn't stressful.
My wife and I have these discussions frequently, and especially when we have particular things that frustrate either one of us. We both respect the other's viewpoint, and we don't tell each other what to do; we say what we think is the problem, then we each provide possible solutions and explanations for how to solve them - often researched in parenting books or papers. (She's a research scientist and I'm a research analyst, so perhaps we go overboard here...) We come to an agreement on how to handle things, and then implement that as best we can - understanding that we're not identical nor are we perfect and so we make mistakes, but we try to stick to what we agreed on, at least until it doesn't seem to work.
We also often discuss specific events after they occur - a while after, so the stress is gone, but not so far after that we don't remember the details (so maybe that night). If we had a particularly difficult issue, we talk about it, and how we could've done better. There are no recriminations here, and the way to get there is mostly for each parent to say how they could've done things better; telling your partner how they screwed up is a recipe for disaster. But if you are both able to talk about how things went, and then ask the partner for suggestions for how to improve next time, it can be very fruitful.
It's also important to ask how you can help her in situations like this. She may appreciate it if you intervene in certain ways; my wife for example appreciates it when I recognize that she's losing her temper, and I find a way to step in and remove one of the children from the situation (We have two 19 months apart, similar to yours). But she doesn't appreciate it when I come in and start telling the kids what to do directly - as I may be not understanding what she's trying to do. We've talked about this several times, and I've learned how I can be helpful instead of just taking over and inserting my judgement for hers.
