The answer to this question depends on what country you are asking about.
In the United States, we have a presidential system where the president is the head of government. In this system, the difference between the president and the government is that the president is one part of the government. In the US, we tend to refer to the entire executive branch (and sometimes to the legislature as well) as the government. This means that the president is similar to the government because he (or someday she) is part of running the country, while he is different from the government because he is only one individual while the government is comprised of many individuals.
There are countries, however, where the president is not the head of government. In such countries, the president is the head of state, but not head of government. Ireland is such a country, as is Germany. In these countries, the president is really not part of the government. The term “government” typically refers to the cabinet ministers and the head of government (often called a prime minister). The president and the government, then, are very different because the president does not play a real part in the day-to-day running of the country. The president and the government are similar in that they both represent the country as a whole in some ways, but the president is really just a symbolic leader while the government is the body that actually runs the country.