I'm all for being contrarian, but Chris Good is wrong about the importance of the president's budget. Yes, it's true that the budget is only a proposal, and that it's Congress that has to pass a budget. It's also true that the "budget" portion of the budget process may turn out to be only tangentially related, in many years, to actual tax and spending changes. It's also true, as Ezra Klein points out, that a lot of things that go into the budget aren't really the "president's" ideas; they're the residue of decades of policy decisions and economic performance.
All that said...yes, the president's budget is actually a significant piece of work. Presidents, or at least those actually interested in governing (Obama clearly qualifies), don't just fill in numbers to make it look good. The items in the president's budget reflect the administration's position, which is often arrived at through tough negotiations between agencies, relevant interest groups, and the White House. Especially during times of unified government, these positions tend to be a good indication of how the final results will be, months later.
Now, that's not always true; even when one party controls the White House, the House, and the Senate...wait, let me put that better. Parties don't "control." Even when the president, the majority of Senators, and the majority of Members of the House are all affiliated with the same party, there are still differences in their constituencies, which will have effects on policy bargaining. You know about this: Senators are probably more relatively more responsive to rural constituencies, presidents may pay back their good friends in Iowa and New Hampshire...sometimes it matters who the holds the chairs of the relevant committees and subcommittees. Still, the White House remains a big influence on what happens in Congress. Moreover, the outcomes of the White House budget process tend to reflect which programs and which groups are in and out of favor not just with the president, but with other same-party politicians, and so the White House budget tends to predict Congressional action because a lot of the same influences are brought to bear on both.
Lat thing: a lot of what matters isn't the top line overall spending and deficit numbers, but the individual program suggestions, whether it's the possible end of the manned return to the moon or reduced energy subsidies. And I know about those two because they reached the level of news today (most likely because there are going to be fights about them); there are dozens of other choices, many of which most of us won't ever hear about, because our news media doesn't cover policy enough. But each of the policy choices has winners and losers, affecting people's lives, and so these choices matter quite a bit. Even if the president's budget is just one step in the process.