Albuquerque, New Mexico, USA has a hundred times the population of Alburquerque, Badajoz, Spain.
The one in the USA is famous mainly as a joke in a Bugs Bunny cartoon.
I Don’t know anything about the one in Spain.
Arlington, Texas, is more populous than Arlington, Virginia, but Virginia is famous for the cemetery and the Pentagon. They weren’t named after each other, but both were named after Robert E. Lee’s home, Arlington House. Technically, Arlington, Virginia, is a geographically small county, not a city. But it is often treated as if it were a city, and there have been proposals to make it official.
San Jose, California and San Jose, Costa Rica, perhaps? One is the epicenter of Silicon Valley, the other is the nation’s capital and largest city. The one in Costa Rica is significantly less populated, but has a big enough tourism draw that I’d argue it punches above its weight in terms of importance. Its airport has a fairly high number of flights from Europe for a city its size. Meanwhile the one in California gets overshadowed by San Francisco which makes it feel less important than a city of its size should.
Let’s not forget the Springfield in Oregon, home of Bart Simpson. 
There are other Monterey’s in the US, all rather small. None were named after the one in California, but rather after the Battle of Monterrey in the Mexican-American War. However, they all are spelled like the one in California (with one R) and the one in Mexico with two.
And something about a parade and game. I assume you mean “unusual” in that the Texas city was named after one further to the west. That is indeed uncommon for North American cities. However, note that Pasedena, Newfoundland is also named after Pasedena CA.
Another possible answer to the OP is Abilene TX, which was named after the town in Kansas. The one in Kansas is smaller, but better known if only because it’s the hometown of Dwight Eisenhower. So those balance each other out, I think.
Cambridge, England (population 125,000) and Cambridge, Massachusetts (105,000). But if you want to get fussy, the Massachusetts city is named after the English University, not directly after the English city.
Both cities called Memphis are important, but in very different ways. The city in Egypt is now uninhabited but it remains important to history and archaeology. The the city in Tennessee is home 650,000 residents and well known for its ties to modern American music. Inside the US, the Tennessee city is probably more famous, but worldwide maybe ancient Memphis is.
I can’t think of any other ancient cities whose modern namesakes approximate the same level of fame as the original. For instance, if you mention Troy out of context, I would assume you’re talking about the abandoned ancient city near the Dardanelles, not the small city of 50,000 in New York.
And one little old lady.
How about Philadelphia? The American city was named after an ancient city in the Middle East; that city still exists, except it’s now known as Amman, the capital of Jordan.
Yes, though Cartagena de Indias could arguably be of “greater importance” – it certainly is a lot larger, and is a UNESCO World Heritage site.
I second the OP’s nomination of Birmingham, England, and Birmingham, Alabama. The latter was named after the former, specifically because, like the city in England, it was becoming an industrial center, particularly in the steel industry.
There were three cities in the ancient world named Philadelphia, but the modern city was not named for any of them. One of those three is now known as Amman, the other two, one of which is mentioned in the Bible, were places in what is now Turkey. So what city was Philly named for? None at all; it was named for its meaning in Greek: City of Brotherly Love.
I don’t know about that. When I traveled in Europe and told folks I was from Memphis, no one ever mentioned Egypt. But every single one of them asked about Elvis. I think he made Memphis TN more famous worldwide than ancient Memphis for everyone except history buffs.
I was thinking Portsmouth, NH and Portsmouth, UK.
Wiki says that UK is a lot bigger in population than I thought
Stratford-upon-Avon in England is renowned as the birthplace of Shakespeare and the home of the Royal Shakespeare Theatre.
Stratford, Ontario is named after Stratford-upon-Avon, and is also a theatrical center; it’s the home of the Stratford Festival (originally the Stratford Shakespeare Festival), one of the greatest summer theater festivals in the world.
Yep. And if you live in one of them, make sure you get on the correct return flight: is it RCH or RST?
I was once in a cab in Rochester, NY and asked the driver if he’d ever had a passenger ask to take him to the Mayo Clinic. He said, yes, but had no idea why, since he did not know that the Mayo Clinic was in Rochester, MN.
Both US Rochesters have populations of around 200,000, 3x bigger than the original Rochester in Kent, England, which dates back to Roman times.
Rochester NY is named for a Col. Nathaniel Rochester; Rochester MN was named for Rochester NY.
Vancouver, BC and Vancouver, WA are about of equal importance, and close enough to each other that you could potentially wind up in some trouble if you didn’t specify which one you were headed for.
When Vancouver BC hosted the Winter Olympics, there were a number of cases of people making reservations at Vancouver WA hotels by mistake. I can’t remember if anyone showed up for them, expecting to be near the Games. Probably happened, but the commercial airport for Vancouver WA is Portland Int’l, not Vancouver Int’l, so hopefully that would give people a clue they weren’t going to the right place.
There’s Greenville, NC (population 90,000) and Greenville, SC (population 64,000). The only reason I know about these two is because I knew a guy who drove to the wrong one for a job interview. They’re around 350 miles apart so as you can imagine things did not go well. This was better than 30 years ago so no google maps to help out.