In 3.x, some spells state that they "counter and dispel" other specific spells. Does a spell used in either of these ways still have it's normal effects? Does this depend on the specific 3.x system in use?
When a caster counterspells using a specific spell that counters a specific opposed spell, the specific spell has no effect beyond countering the specific opposed spell...
Dungeons and Dragons 3.5 on Counterspells says, in part, that
It is possible to cast any spell as a counterspell. By doing so, you are using the spell’s energy to disrupt the casting of the same spell by another character. [...] [In addition to other steps unimportant for our purposes, t]o complete the [counterspell] action, you must then cast the correct spell. As a general rule, a spell can only counter itself. If you are able to cast the same spell and you have it prepared (if you prepare spells), you cast it, altering it slightly to create a counterspell effect. If the target is within range, both spells automatically negate each other with no other results. [...] Some spells specifically counter each other, especially when they have diametrically opposed effects.
Emphasis mine. That is, instead of using the same spell or the spell dispel magic et al. to counterspell, the counterspelling caster employs the spell that specifically counters the identified spell. The spell the counterspelling caster uses to counterspell only counters the identified spell; the spell has no other effect. Although Pathfinder makes some diction and syntax changes elsewhere in Counterspell, this text remains unchanged from Dungeons and Dragons 3.5.
...But using a specific spell to dispel or negate an opposed specific spell is addressed only in FAQs
From the Dungeons and Dragons, Third Edition FAQ:
When an opposite spell is used to negate an effect that is already in place (such as using slow spell to counter and dispel a haste spell), is the success of the dispel automatic, or is a level check required? Does the subject or the caster of the spell being countered and dispelled get a saving throw?
Two opposite spells simply negate each other. No dispel check is required, no saving throw is allowed, and spell resistance does not apply.
I was unable to find a direct link to the Dungeons and Dragons, Third Edition FAQ (my search results occluded by the Dungeons and Dragons 3.5 FAQ), but this text is both quoted and referenced in several related threads, such as this 2004 ENWorld thread. Dungeons and Dragons, Third Edition material not updated for Dungeons and Dragons 3.5 remains legal in Dungeons and Dragons 3.5 play, with minor modifications made by the DM (DMG 6). As this particular ruling wasn't updated, it should remain legal... or, at least, as legal as the FAQ gets. (Some say the 3.5 FAQ anyway has issues of its own.) Barring miracles, this ruling might be as close to official as will ever be available for D&D 3.5.
The Pathfinder Core Rulebook FAQ addresses the issue in much the same way:
If I use a "diametrically opposed" spell to counter or dispel another spell (bless vs. bane, haste vs. slow, and so on), does my spell have any effect other than dispelling?
It depends on whether you are using the spell as a counterspell or as a dispel.
If used as a counterspell, your spell has no effect other than to counter the target spell. If used as a dispel, there may be "spillover" from your spell or the target spell, depending on whether you affect more or fewer targets than the opponent's spell.
Counterspell Example: You are a 5th-level wizard, your opponent is a 6th-level sorcerer. On your turn, you ready an action to counterspell. The sorcerer begins to cast slow. You succeed at the Spellcraft check to identify the spell and cast haste as a counterspell against it. Your haste counters the slow, and neither spell has any effect.
Dispel Example: You are a 5th-level wizard, your opponent is a 6th-level sorcerer. On her turn, the sorcerer casts slow and targets 6 of your allies; all 6 of them fail their saves and are slowed. On your turn, you cast haste and target 5 of your allies; this automatically dispels (no caster level check needed) the slow spell on those allies, leaving them without the effect of slow or haste (your 6th ally is still affected by slow). Note that this does not merely suppress the slow effect for the duration of your haste—the effect is completely dispelled on those 5 allies. Note that it doesn't matter if the target would normally get a saving throw or spell resistance to negate or avoid the spell used to dispel (such as casting slow to dispel an already-caste haste); to speed up gameplay and prevent lopsided applications of this sort of dispelling, the "diametrically opposed" spell automatically dispels its opposite, regardless of the desires of the creature affected by the opposite.
Pathfinder uses its FAQ often as a substitute for errata, therefore being official rules, and, for example, used in organized play. As a still-published game rooted in Dungeons and Dragons 3.5 material, a DM could do worse than house rule his Dungeons and Dragons 3.5 campaign's counter and dispel language to function like such language functions in Pathfinder.
There are 2 different actions here: Counterspelling and Dispelling.
On Counterspelling, from the Pathfinder Core rulebook:
If the target is within range, both spells automatically negate each other with no other results.
So no, these spells do not have their normal effect, they are both negated.
On Dispelling, there is some confusion, but from RMorrisey's answers to this question, once the second spell successfully effects the target of the first spell, the first spell is removed and the second spell has no further effect. They negate each other, leaving whatever would 'naturally' be there.