Please explain whether in German the phrase in die Stadt fahren is the same as in das Zentrum fahren. My course’s exam today had shown both these places as the same. I had thought Stadt and Zentrum are different. Driving in the city should be different from driving to the centre of the city. Am I right?
The word Zentrum is mostly limited to written texts. People would rather say Innenstadt or Stadtmitte. When coming from outside town, there's little reason to assume you would go to the suburbs when saying Ich fahre in die Stadt. If you do, you would use the name of the particular suburb instead. So Stadt replaces Innenstadt here.
Es hängt von Stadt und den Umständen ab.
Berlin hat beispielsweise 2 Zentren, eins im Westen, eins im Osten.
Dann hängt es davon ab, wo der Sprecher ist. Wenn er außerhalb der Stadt ist, dann kann das Überqueren der Stadtgrenze als in die Stadt fahren verstanden werden. Wohnt man eh in der Stadt, aber in einer Wohnsiedlung am Stadtrand, kann in die Stadt ein Synonym fürs Zentrum sein.
Insbesondere kann die Situation so sein, dass sich aus ihr ergibt, was gemeint ist, etwa wenn eine Familie zum Einkaufen entweder in den Laden außerhalb der Stadt, oder ins Geschäft in der Stadtmitte fährt, und jemand, von dem man weiß, er will einkaufen, sagt, dass er in die Stadt fährt.
Es sind aber auch Siedlungen denkbar, die ein Zentrum haben, aber keine Stadt sind.
Some answers here claim that ins Zentrum fahren be unidiomatic or never heard. To get that out of the way: while it is generally true it is not always true. Depending on the city in question and the people saying it, the more common phrase for going to the centre can indeed be ins Zentrum fahren. I will admit, though, that it is more common in most cities to say Innenstadt, Stadtmitte — or maybe you use a name of an actual borough or a prominent place in the centre instead.
Now that that has been said let’s look at the differences between the two. No matter where you’re coming from, ins Zentrum fahren will always go to the centre. That means a couple of things:
for cities like Salzgitter, which do not have an established city centre, you cannot say ins Zentrum fahren
if you are coming from way outside the city and you are just going to a suburb (which already belongs to the city but is away from the centre) you again cannot use it.
What exactly is meant by in die Stadt fahren depends a lot on who says it. going from more central to less central:
if you already live in the centre you probably won’t use it because you’re already there. However, you may say things like:
Ich wohne in der Stadt.
If you live in a borough of the city that is not exactly the centre (e.g. Munich Pasing), you would most likely use it exactly in the sense of ins Zentrum fahren — i.e. you’re going to visit the city centre.
If you live just outside the city (often, but not always delimited by political boundaries; however, suburbs that are more like outside villages count here, too), it can either mean going into the city at all (i.e. going any place in the city’s boundaries) or as above going to the city centre.
And finally, if you live far out of town (at least ten minutes drive) you would use it for every trip inside the city’s boundaries, even if you’re just visiting a friend who lives on the very outskirts.
For the second group, both phrases are equal, for the third group, they are practically equal. The first won’t use either and for the fourth they have different meanings (akin to the difference between city and city centre)
‘Ins Zentrum fahren’ has the connotations of to go "downtown."
‘In die Stadt fahren’ technically means "to go into the city," so you have a point there.
But its current connotation is also "to go downtown." For our purposes they are the same, nowadays. That might not have been true 50 or 100 years ago.