It depends on the spell.
Restorative Touch will remove one minor condition of your choice. If this condition was gained as an aftereffect of the spell with an instantaneous duration (like Snowball or Frigid Touch), alchemist's Frost Bomb, Staggering Critical feat or the like, the condition is removed.
However, Slow is an ongoing spell (Duration: 1 round/level). Its magical energies linger for several rounds, slowing the creatures for the duration, therefore the staggered condition will just be immediately reapplied.
Let's start with spell durations as outlined in Magic section of CRB:
Many durations are measured in rounds, minutes, hours, or other increments. When the time is up, the magic goes away and the spell ends. If a spell’s duration is variable, the duration is rolled secretly so the caster doesn’t know how long the spell will last.
The spell energy comes and goes the instant the spell is cast, though the consequences might be long-lasting.
Note the emphasis. It strongly implies that the effect is tied to the spell energy being in place; as soon as it's exhausted, the effect goes away.
Let's consider a following scenario. Our 4th level party standing off against evil wizard Wendy. She wins initiative and casts Hypnotism on the party's barbarian Bob, who fails his will save and is now fascinated for, say, 5 rounds (average of 2d4). Party sorcerer Stella counters that by casting Suppress charms and compulsions, breaking Bob's fascination, who is now set on murdering Wendy for the insult and charges the poor girl. However, he does not notice the evil ranger Ralph, who shoots an arrow at Stella the next round, breaking her concentration. With Suppress charms and compulsions now gone, Bob again falls under the ongoing energies of Hypnotism and is fascinated again.
Stella is now furious, so she pulls out a scroll of Dispel Magic and succeeds on UMD check to cast it, and also succeeds a caster level check to dispell Hypnotism. With spell energies now gone, Bob is no longer fascinated and can rip Wendy a new one.
Now, let's suppose that Wendy still not done and casts Frigid Touch and crits on a touch attack, staggering Bob for a full minute. Now, note that unlike Hypnotism, duration of Frigid Touch is instantaneous, so spell's energies are gone and Stella can't dispel the effect. Thankfully, cleric Charlie finally catches up to the rest of the party and uses his Restorative Touch on Bob, ending his stunned condition.
Note that how fascination comes and goes with spell energies being suppressed, restored and finally dismissed. Same effect would happen if, say, instead of Stella casting Suppress Charms and Compulsions, rogue Rob would bring an artifact stone with permanent Antimagic Field next to Bob, and then him being bull rushed out of range by Ralph's ram animal companion.
In short, to remove an effect caused by ongoing magic, you deal with magic itself, not the effect. Removing an aftereffect of magic use, you deal with the effect itself.