The one on the right, a yucca palm, has a very wobbly stem. This should be much straighter and thicker. This stem is probably a sympton of stunted growth so I guess the plant was under stress when it grew this. There can be multiple causes for this. Also, the leaves shouldn't be hanging down like that but be more outstretched, and I guess this is another symptom of the same problem. A palm that size also deserves a bigger pot. I recommend you repot it in a bigger pot with adequate drainage (about 1 cm of fine gravel at the bottom) and then soil. You can get special palm soil at garden stores so use that if you can find it or otherwise normal planting soil will do. When removing the plant from the old pot, try to loosen the roots up a bit. They've probably been growing around in circles for years and used up all the good soil, so try to spread that a bit so they get into good soil more quickly. On your first watering, be very generous and maybe place the pot in the sink so the water can drain out. After that rewater about once a week, and use fertilizer sticks in the summer (the green ones, not the red ones). Also watch the leaves. If they are attracting dust, clean them once in a while with a moist cloth. If there are insects or anything like that, spray them. Keep the plant away from draughts, and also keep away from direct heat sources so don't place this plant next to your radiator.

Whereas the existing leaves won't improve, you'll probably find that any new leaves it grows will be larger and straighter and won't flop so much. Also, the stems will start growing straighter, or sometimes (rarely) a fresh stronger stem will branch out from a lower point and ultimately replace the other ones..