Beowulf and Odysseus both faced many hardships, trials, and injustices in their stories. When they were faced with those trials, they always had a choice in how to react to them. They both had similar and different ways of responding to these, whether in a heroic manner or not.
To begin with The Oddysey, after the Trojan war, all the surviving warriors were able to return to their homes and families, but Odysseus was stuck imprisoned on Calypsos island. After seven years he finally manages to get away only to continue feeling the Wrath of Poseidon who was mad at him because he took out the eye of one of his sons, Polyphemus. It took Odysseus twenty years between when he left for the war until he finally made it make to his home in Ithaca. Now Compared to Beowulf, Odysseus wasn’t as infallible or heroic as him, he was super smart.
He was able to think of plans and execute them in ways that would eventually end up in him getting his way. Granted, he was a bit lucky in some areas, but for the most part, he would deal with external problems with his brain. A few examples of this could include him pretending to be into some Women to earn their favor or the story of him Telling Polyphemus that his name was ‘Nobody’ so that when he and his crewmates blinded him with a pole, they wouldn’t draw any attention to themselves when he would yell that ‘Nobody’ was the one that blinded him. Odysseus did have to show feats of physical strength, I don’t want to discount that, but overall, as I’ve stated, he was able to face his hardships by outsmarting his opponents and leading others into overcoming them.
Comparatively, Beowulf would face his trials, injustices, and hardships by his infallibility and strength. He would face his struggles head-on, running in to fight or battle whatever it is that challenges him. He battled and fought many people and monsters and such and (almost) always came out on top. I think to determine whether or not his actions were heroic or not is dependent on what the intent of his heart was in doing them.
Even though he did do a lot of good, often times he would do it to benefit himself in such ways to gain glory or fame. Later on in the story, he begins to have better motives in his acts, like when the slave finds and takes a gem from a tower and causes a Dragon to wreak havoc on the surrounding areas, Beowulf deemed it his responsibility to stop and kill the dragon so he could protect his people. He ended up dying because of this, but I think that makes it even more of a heroic act because he was willing to accept the fact that he could and ultimately would die.
While both Odysseus and Beowulf approached their external problems differently, in doing the things they did, they both ended up making it back home at the end. They both typically did what they had to do and nothing more.
Overall, if I were to compare Odysseus and Beowulf to determine whether or not their acts were heroic or not, even though they both had different motives at times and they did things differently, I would have to say that they were both heroes in the ways that they faced their injustices and trials.