As depicted in Beowulf, compare Beowulf's fight with Grendel with his fight with Grendel's mother in order to define which seems more dangerous.
The epic battles in Beowulf prove to be very different in regards to difficulty for the hero Beowulf. Beowulf's fight with Grendel was one which he was prepared for. Being the true hero he was, Beowulf chose to enter into the battle in the same way Grendel would (without weapons). In this sense, Beowulf knew that he would be required to fight Grendel with his hands and strength alone.
In Beowulf's battle with Grendel's mother, Beowulf took Hrunting (the heirloom sword of Unferth's family). Upon beginning battle with Grendel's mother, Beowulf soon finds out that the weapon is useless against the monster.
In both battles, Beowulf reigns as the victor. This said, the battles are very different. The battle with Grendel proves to be easier than the battle with his mother. For example, Beowulf must only fight Grendel. There are no other obstacles that he must overcome. In the battle with Grendel's mother, Beowulf must face off against sea-beasts which dwell in her watery lair.
One last difference between the battles was the lead-up to the battles themselves. For the battle with Grendel, all Beowulf needed to do was lay back and wait for Grendel to appear. On the other hand, he was required to spend "most of his day" traversing the watery dominion in order to reach Grendel's mother.