They can both refer to physical touch, but "connect" has a sense of sureness and "contact" is often used when you find someone and start communicating with that person. Look for sentences that use each word - you'll see the grammatical pattern is different as well.

Both words have many meanings and senses. But the similarity is about other people. When you "connect with" someone, it can be either establish communications or to find a common interest. eg At the meeting today I was able to connect with the new supplier representative. or During lunch today, I was really able to connect with our new co-worker. I think we will be able to get along well together. contact just means to communicate with. eg I need to contact the landlord because the heat in the apartment is not working. but it can also be a noun. eg I have a small circle of friends - just 50 contacts in my phone.