Okay, so this is a difficult one, and we have no satisfying answers right now. BUT, I can try to give a bit of the neuroscience background that could possibly underlie this.

I'm curious as to what happens when someone makes new connections. Does anything change in the person's "mind"?

Basically every neuroscientist is going to agree that the mind arises from the brain and so any changes to the "mind" entail previous changes to the brain. The hard bit is asking about whether the causality also goes the other way around -- do physical changes to the brain entail later changes to the mind? This depends on how you define the mind and is probably an unanswerable question as of right now.

Are there any physical changes to the brain? New wrinkles to the brain's surface?

Yes! Learning something new, having some insight, making connections between related things (as in your example) all come with neuronal learning. Neuronal learning is accompanied by physical/chemical changes to synapses, which are basically the communication sites between neurons. New synapses can form, synapses can become larger or closer together, etc. But this isn't going to result in new wrinkles in the brain's surface; those wrinkles, also known as sulci, are what we call macroanatomical structures: they're made up of so many millions of neurons that nothing drastic enough to change them happens after initial development as a baby.

I guess a simpler analogy is humor; what happens to the brain when someone finds similarities between two apparently dissimilar things?

No one knows. But to get more insight into this, let's think about this question by starting with the distributed nature of information storage in the brain. It's not the case, for example, that memories exist as separate, boxed-off entities with their own dedicated neurons for each memory. Instead, networks of neurons distributed across disparate regions of the brain contribute to firing patterns that "represent" a memory. So whenever you're doing some kind of thinking activity, huge regions of the brain are firing with largely overlapping areas. So when you're thinking about something, certain bits of other networks become active. This is where my speculation comes in: it's possible that when some subset of overlapping networks is activated, this could trigger full recall of another subset of those networks, causing the experience of realizing a similarity you had never thought of before. But again, this is speculation and we really have no idea.