One is that he's highly unlikely to succeed, and presidents don't look for opportunities to lose. Getting a constitutional amendment requires mobilizing a strong national consensus on a particular issue. It requires persuading each house of Congress to muster a two-thirds vote in favor of a specific proposal. And it requires ratification by three-quarters of the states.
Even if the amendment had a plausible chance of passage, it would probably take longer to achieve than a president's tenure in office. He would use up a lot of precious time and political capital that could be devoted to more achievable goals, with more immediate rewards. All this would get in the way of pragmatic, incremental changes.
It may make sense for pro-life groups to hold out hope of a constitutional amendment, since they are in the fight for the long haul. It makes much less sense for a president, who will be gone in four years or eight.
In some cases, a proposed amendment would give members of Congress an excuse to put off serious legislative action. You can prove you're serious about the government debt by voting to raise taxes or cut spending. But that means antagonizing voters who would have to endure unwelcome sacrifices.
So why bother? It's much easier to demonstrate your fiscal conservatism by voting for a balanced-budget amendment -- while opposing the actual fiscal changes it would require. Anti-abortion candidates can endorse an amendment without much risk, since pro-choice voters know it's not going to pass.
When a presidential candidate vows to amend the Constitution, he may be doing any number of things: dodging a tough issue, pandering to a bloc of voters or trying to sound bold. What he is not doing is telling the truth.