If you had wounded legs, then the potion of speed will heal them. (Normally, you can heal your legs just by waiting some turns, so do not waste a potion of speed to heal them except in emergency!) Otherwise, an uncursed potion will haste you 100 to 109 turns like the spell. A cursed potion will haste you only 40 to 49 turns, but a blessed potion will last 160 to 169 turns.
Ultimately though, if you can find speed boots, then the spell and the potion become redundant. A Wizard or other good magic user with intrinsic speed might be able to retain the haste self spell while wearing different boots. Meanwhile, the potion of speed becomes a candidate for alchemy or dilution to make holy water.
The wand is more useful and important. You can use the wand of speed monster to give permanent intrinsic speed to both yourself and your pets. Hasting your pets allows them to hit more often in melee, unless they already have 12 speed; your pets will be stronger because melee is their primary activity (other than stealing from shops and curse-testing).
Haste self is also a monster spell; when a monster casts this, they become fast. It produces the message "Foo is suddenly moving faster."