Third Class medical license, what changed, if anything?
Short answer: politics killed the "driver's license medical" idea for private pilots, but there are still reasons why BasicMed has significant advantages for some pilots.



The literal answer to your question is "nothing": the third-class medical hasn't changed at all. What you're referring to is an additional option called BasicMed, which you can hold instead of (for some operations) or even in addition to a third-class medical.

I won't get into the details of BasicMed itself (the regulations are in 14 CFR 68), but AOPA has a lot of great information about it. Instead, I'll focus on your "why is this any better than a third-class medical?" question, which seems to be quite a common one right now.

In summary, if you're under 40 and healthy then BasicMed may not seem very interesting. But if you're over 40 and especially if you currently have a Special Issuance then BasicMed could be a much easier and cheaper way to fly.

As you said, as a new pilot you still have to pass a third-class medical once so let's assume that you've done that successfully. It's certainly true that some people aren't happy with that requirement because they were hoping for a "driver's license medical" where a private pilot could fly without any medical certificate, as sport pilots can do. That idea didn't survive the political process, however. If you want to fly without any medical you still have gliders, ultralights or LSAs.

But when it comes time to renew, here are some reasons that you might consider BasicMed (most of this information comes from the AOPA FAQs):


If you're over 40, you need to renew a third-class medical every two years, but you only need to see a doctor every 4 years under BasicMed
Special Issuances still exist under BasicMed but for a more limited number of conditions than for a regular medical. Practically, that means you may be able to avoid expensive tests that the FAA requires for a third-class. This is a huge benefit for some pilots because it can save a lot of time and money (health insurance typically doesn't cover 'unnecessary' tests).
Any state-licensed physician can sign off on BasicMed, but only an AME can sign off an FAA medical. That means in theory it should be easier and cheaper to find a doctor for BasicMed. In practice people's experiences have been mixed on this, but BasicMed is still very new and it's not surprising that some doctors may not be comfortable with it.
Because the BasicMed medical questionnaire is effectively a general physical exam, some people have successfully combined it with an annual checkup paid for by their health insurance


There may be other reasons, but those are some common ones I've read about. If you're a member, the AOPA forums have a lot of discussions about individual experiences.