The differences between these two are much more important than the similarities. In fact, it is often said that political parties and interest groups are in competition with one another for power.
There are some similarities between these two. The main one is that they both want to get the government to do certain things. The NRA and the Republican Party, for example, both want the government to do less in the way of regulating people’s right to bear arms.
However, this is where the similarity ends. The major difference between the two is that the political parties care about many issues while interest groups only care about one. Using our previous example, it might be in the Republican Party’s interest to downplay issues of gun rights. The party might be able to attract broader support if it did not seem so dogmatic on this issue. By contrast, the NRA cares only about gun rights. It does not particularly care if the Republican Party expands its base just as long as only pro-gun people are elected to Congress.
Political parties try to create and hold together broad coalitions. They want to be able to govern effectively over a range of issues. Interest groups do not care about broad coalitions. They only care about winning on their particular issue.
The similarities between interest groups and political parties are broad. Both interest groups and political parties try to shape political policies. Both try to protect and promote the causes they support. Beyond that, they differ quite a bit.
An interest group is a group that shares a common interest. (That sounds pretty obvious.) These groups can be formal, informal, or both. For example, you could have a large, informal interest group made up of several formal groups, as well as unaffiliated individuals. The environmental movement is a large, diverse interest group. It contains dedicated individuals as well as quite distinct organizations. The main thing that unifies an interest group is commitment to a specific cause or interest. They might actually disagree about what form support for that cause would take. Interest groups might donate to politicians, or support specific legislation, but they are not usually a formal part of the government.
Where an interest group focuses on specific interests, political parties often represent an array of interests. Ideally, these parties are unified, but that's not always the case. Political parties share a history and an ideology.
As an example, consider the Libertarian Party. It is unified by its support of individual liberty and the free market and opposition to government action. Many different interests groups might support this, interest groups that would otherwise have little to do with one another. For example, someone who supports drug legalization (and is part of that interest group) might find him or herself beside an abortion rights activist (another interest group) in supporting the Libertarian Party (a political party).