Why was the government expenditures of the Roman Empire very low compared to modern governments?

The methodological caveats listed by @Alex are all true. I'd also add that there was no notion of state credit so everything had to be financed out of revenues. And of course, the very notion of what government is responsible for was quite different in antiquity - the ancients were very far from the idea of a welfare state. Perhaps early 19th century England or France would be a better model for comparison.

While I agree with the other answer givers that you're unlikely to find reliable information on Roman imperial spending budgets, I think we can usefully work back from modern spending figures to show the disparity.

For example, below are some of the major spending areas of the United States budget, and a note about Roman equivalence:

1. Social Security is a huge portion of US federal spending ( about 20%). The Romans had no state care for the elderly, who would have lived with family.
2. State funded healthcare (Medicare and Medicaid, as well as emergency room treatments) are another 20% of the budget, and didn't exist in Roman times.
3. Public Works and infrastructure is about 4% of US federal government spending, but there's more money supplied by the individual states. This is probably about equivalent to the Romans, who built roads, aqueducts, public baths and more in most of the empire.
4. Defence spending is about 25% of the US federal budget, and was a much smaller component of the Roman budget. However, I'm not sure the militaries are comparable. Obviously, the Romans didn't have to pay for the upkeep of a nuclear arsenal, submarines or an airforce, which are all incredibly expensive in terms of equipment costs. Additionally, Roman soldiers were paid partially in spoils of war, which were also a form of income for the empire. Obviously, this isn't something the US government does.

Anyway, that was a quick comparison to the US. Note that even in modern times, the US spends more on defence than the next 26 countries combined. So perhaps the defence budget of the UK, France or Australia is more comparable to Roman numbers.

Addition: I have now looked up the UK's 2014 budget figures, and defence spending is on the order of 5% of spending, which probably puts it at about 2% of GDP.

First of all, I would not trust Wikipedia numbers about Roman Empire. Roman empire existed for 4 centuries, and the things did not stay unchanged. Second, we have no reliable statistics for most periods. Even the population of the Empire in various periods is not clear, and estimates widely vary. Third, you cannot compare ancient economy with modern economy, it is simply ridiculous. Some portion of the Roman economy was based on slave labor, for example. And again, we do not know what portion, and how many slaves there were. Slaves did not pay tax. Fourth, most European countries nowadays spend less than 2.5% of their GDP on their military. Including NATO members. Fifth: the government funds in the Roman empire were not only spend on the army; they did huge construction works everywhere in Europe (which you can see even now), then in various periods the population of Rome itself not only did not pay taxes but was supported at the government expense.

With all this, it is of course true that ancient societies could not spend as large part of GDP on government expenses as modern ones, simply because the labor productivity is incomparable. For most of the history most people worked just for subsistence. It is only surplus production besides the subsistence, which can be used on all other expenses, including military.

And you don't have to use the Roman empire for comparison. Compare the part of GDP which was collected in taxes in the late 19-th century US with the modern numbers:-)