WASHINGTON—President Barack Obama and Japanese Prime Minister Shinzo Abe touted reinvigorated security ties and progress on a trade accord as the two leaders vowed to work together on issues ranging from maritime disputes to cyber threats.

After Tuesday’s White House meeting, though, Mr. Abe faces a tougher venue Wednesday when he addresses a joint session of Congress and campaigns for a sweeping, 12-nation trade agreement. On Capitol Hill, Mr. Abe will encounter strong Democratic resistance to the Trans-Pacific Partnership...