I've had a chance to look at the paper. So there were two tasks, one which contrasted global and local information processing when comparing figures based on global or local features, and another which measured three aspects of attentional control: alerting, orienting, and interference, in which the primary measure of interest was response time. There was no contrast between an emotional arousal treatment and a non-aroused treatment, as I had thought. Instead, this contrasts with previous literature that did not compare caffeine and theanine under conditions of emotional arousal.

For the first task, they found that in the caffeine treatment subjects were more likely to pay attention to global features when rating similarity relative to placebo, and that in the theanine treatment, they were more likely to pay attention to local features.

In the second task, they found no significant differences in alerting and orienting, so I won't discuss these. For executive control (interference resolution), there were differences. This task is a flanker task. They have to indicate the direction an arrow is facing when flanked by arrows that were either pointing the same direction (congruent), pointing the different direction (incongruent) or neither (neutral). They then computed difference scores by subtracting the mean response time from congruent trials from mean response times from incongruent trials. Ordinarily, incongruent trials are harder, so response times will be slower than in the congruent condition. To quote from the article, which substantiates what I said about how these difference scores are generally interpreted, "for executive control, higher difference scores indicate poorer functioning." So what were the results? Relative to placebo, in the theanine treatment subjects had significantly larger difference scores, which the authors would interpret as poorer executive functioning. Relative to placebo, in the the caffeine treatment, subjects had significantly smaller difference scores, which, again, the authors would treat has better executive functioning.

To reiterate what I said before: difference scores lose information. The authors do not show the raw mean response times by condition. So just because the difference scores were greater in theanine relative to placebo than in caffeine relative to placebo does not, in my opinion, unambiguously tells us about performance. For example, it could be that in the caffeine treatment, people get slower in congruent trials and the incongruent response times remained the same.

They also collected subjective ratings of mood. It's worth quoting them on that:

"Caffeine consumption, both with and without theanine, elevated feelings of tension, depression, anger, confusion, and total mood disturbance. In the absence of theanine, caffeine also enhanced feelings of vigor and depression. Such findings are consistent with previous studies showing that caffeine reliably increases self-reported
arousal measures across a range of doses (e.g., 100–400 mg caffeine) (Addicott and Laurienti 2009; Mahoney et al. 2011; Smith et al. 2006). Thus, under heightened emotional arousal, the mood effects of caffeine and theanine appear to be driven by caffeine."

So is caffeine or theanine better? Well the experimental tasks are really hard to interpret. I guess if you are doing something that requires a lot of control in a distracting context (say landing a plane, when your co-pilot won't stop talking about what they did in Morocco) in an emotionally stressful situation (your spouse just called to tell you that they want a divorce), maybe caffeine would be better than theanine ... for landing the plane at least. But you might get more depressed. (this last paragraph is mostly a bit of dark humor, don't take it too seriously)