They are very much related, but they are not the same god. They are two expressions of a sky god descended from an early form in what was apparently a proto-Indo-European pantheon. Even their names are etymologically connected. Jove Pater - father Jove - leaves us with Jove, which is a cognate of Zeus - and deus - meaning, simply, 'god' or originally 'sky'. This is also related to Twi/Tyr of the Northern pantheon. As Indo-European languages, pantheon, and people (to a certain extent) migrated into western Europe, there was drift in language and traditions, so that eventually we arrive at a Jupiter and a Zeus who were similar and yet distinct.

With Roman expansion, Roman authorities often looked for these sorts of similarities because they believed they were likely to find local manifestations of their gods (since clearly their gods were the only real gods!) and also because identifying similarities served as a means to bind the diverse Empire together. This was an easy conclusion to reach as Roman authority extended over Greece since the Greek and Roman pantheons had not drifted far too far apart. And then, because Greece had such an impressive body of literature reaching back into antiquity - ancient by Roman standards - Romans were prone to adopt much of what the Greeks had to say about Zeus/Jupiter since their writings had so much authority. All this blurred lines that were already not too distinct because of the common ancestry of the pantheons.

To answer your question simply: they began as the same god, they drifted apart, and then when they encountered one another again, the differences in their manifestations became blurred.

Their names are both derived from the Proto-Indo-European god "Dyauṣ pitṛ," which translates as Sky Father. With the softening of languages over time, it is is pretty clear that Jupiter comes from this combination. Zeus seems to come from the first word (sky) alone.

Again, we can see the connection to this root of sky father with Jupiter and Zeus being presented as gods of the sky and thunder. As a population or culture spreads we can expect to see it evolve locally and temporally. Are they the same god? They have the same roots and many similarities, but the culture in which they exist produces some differences. You can decide where to draw the line.

Side fact: Also related to this root are the words deva, divine, and deus.