Josh Willingham played a small role for the Royals last year. He appeared in 24 games and garnered 86 plate appearances in that time. He hit two home runs for a 113 wRC+ and 0.2 fWAR. Unfortunately, an injury slowed him down near the end, but he did have a crucial pinch-hit appearance in the Wild Card game. Willingham was a right-handed "power" bat off the bench.
Jonny Gomes will probably play a small role for the Royals this year. He probably won't appear in 24 games or get 86 plate appearances. With the division locked up, Gomes doesn't need to do much in the regular season other than not get injured. Gomes was acquired to be a bench bat for the playoffs.
Gomes, like Willingham, is an older (34 years old) righty hitter who mashes lefties. His overall career line of .242/.334/.438 for a 107 wRC+ is hilariously misleading. His career line vs lefties: .275/.378/.483 for a 133 wRC+. His career line vs righties: .221/.304/.408 for an 89 wRC+. His performance against lefties this year is relatively similar to his career levels by wRC+; he's far worse against righties though. Don't look at his line against righties this year. It's gross.
Also like Willingham, Gomes is kind of the "Anti-Royals" player. Gomes hasn't had a single-digit walk rate since 2010 when he played for the Reds. I had no idea he played for the Reds. Willingham's last single-digit walk rate prior to joining the Royals was in 2011 for the Athletics, but that was 9.9 percent. I give him a pass. Both players strike out a ton; Gomes has a career 27 percent K rate, while Willingham's 22.4 percent career K rate belied a large increase in later years to the upper 20s.
The main difference between the two players is (was) Willingham's ability to hit righties. He was much less of a platoon player. Thus, Willingham could offer his skills on a more consistent basis than Gomes will be able to. However, at this point in the season, no player available in a trade will be perfect. While Gomes has his issues with righties, Willingham was a health risk.
Defense doesn't matter. Neither player is (was) meant to stand in the outfield. The similarities are there. High walk, high K, some power. Gomes has thrown some pitches this year though - maybe a trip to the mound isn't out of the question.
As far as Gomes himself, he's mainly a fastball hitter. This year, when he swings at an offspeed pitch or a breaking pitch, Gomes whiffs more than half the time. As a righty, he'll get most of his pitches low and away, but he won't swing at them. His zone profile this year is very interesting - he doesn't swing much. In fact, this year he has the lowest swing rate of his career. The zone at which he swings the most, middle and up within the strike zone, is still at only 65.4 percent. He doesn't care for pitches outside the strike zone (remember the walk rate?), nor does he care for pitches low within the strike zone.
Fortunately, due to his swing tendencies (avoiding anything outside the strike zone and low in general), he avoids putting the ball on the ground more than most. He's not a burner. Unfortunately, his rate of grounders among balls in play is the highest it has ever been. Also unfortunately, Gomes' fly ball distance (per Baseball Heat Maps) is on the decline, as one would expect with an aging player. How will that play at Kauffman?
The Royals need to make sure they severely limit his plate appearances against righties in order to maximize Gomes' performance. Gomes needs to rebound by hitting more fly balls, but his overall plate discipline profile gives him some room to be useful despite a power decline. Gomes is a nice little pickup with potential bench playoff implications, just like Willingham was.