It has been said that the invention of zero was a great leap forward, not only in abstract understanding, but in the ability to introduce place value notation and do computations; computing using Roman numerals was notably clumsy.
Zero is not necessary in order to have a place-value system. Nor is zero necessary (or sufficient) in order to develop sophisticated mathematics. The Mesopotamians had a place value system by, at the latest, 1800 BCE, but had no symbol for zero at that time. Theirs was a sexagesimal (base-60) system. Where we would put a zero, they left a blank space. (But only between digits, and not as a trailing digit, which had the implication that a string of digits defined a number only up to a power of 60 (which could, effectively, be negative). The magnitude of a number had to be inferred from context.) By around 200 BCE, the Mesopotamians had introduced a place-holder zero, but there is no evidence the zero was treated as a stand-alone number.
The Greeks did not use a place value system (although Archimedes devised a special purpose place value system for dealing with very large numbers in The Sand Reckoner) and did not recognize zero as a number. Nevertheless, they accomplished great things in mathematics, including developing a sophisticated axiomatic geometry, proving the irrationality of the square root of 2, proving the infinitude of the primes, developing a sophisticated theory of ratios of incommensurable magnitudes, developing the method of exhaustion and using it to compute the volume of the sphere and the area of a parabolic segment, developing the theory of conic sections, and so on.
The Maya, by 1000 CE (and probably well before), had a place value system that did use zero. Theirs was a vigesimal (or base-20) system. Furthermore, zero was treated as a number: counting started from zero rather than one in their calendar. Their calendar was quite sophisticated and they also had a well-developed astronomy. Nevertheless, it does not appear that their mathematics advanced very far (although it is hard to say for sure, since the majority of their artifacts were destroyed). For example, there is no evidence that they had a decent multiplication algorithm.
The example of China has been addressed in the other answer. The rod numeral/counting board system certainly was a place value system. In that system, an empty place was indicated by leaving a gap rather than by an explicit zero. Many of the problem solutions in classic Chinese mathematical texts were formulated as steps for carrying out a computation on the counting board. To develop something like the Chinese remainder theorem doesn't require a zero. The notion "leaves remainder zero" can always be formulated as "divides evenly". (A similar formulation would have been used in Greek mathematics.) As already noted, the counting board technology had an implicit place-holder zero, but not a full-fledged zero. The counting board technology was sufficiently powerful to enable many great achievements, such as a very accurate estimate of pi.
So what exactly was it about the invention of zero in Western culture that was so useful?
The modern zero was, in fact, invented in India. Knowledge of the Indian system diffused, first to the Islamic world, and later to Europe. Indians knew how to perform arithmetic with zero, and introduced the notion that $1/0$ produces an infinite result.
I think it is clear that efficient calculation systems can be developed without the recognition of an explicit zero. That is not to say that recognition of zero as a number was not extremely important. To give but one example, the theory of roots of polynomials becomes much less cumbersome when zero (and negative numbers) are recognized. Before that happened, the theory involved a complicated case-by-case analysis. The algebraic solution of the cubic and quartic, discovered by del Ferro, Fontana, Cardano, and Ferrari during the Renaissance, as well as earlier work by Omar Khayyam, saw the equations $x^3 = a$, $x^3 + bx = a$, $x^3 = bx + a$, $x^3 + bx^2 + cx = d$, and so on, as different equations, each requiring an individual treatment. The ability to treat such sets of related equations in a uniform manner certainly accelerated progress.