Directly addressing your question, the only skills that are similar enough to be an outright waste to invest in multiple are generally the combat skills.
- It's almost always better to focus on a single armour type, to get matching set bonuses, so as you mentioned, investing in both is unnecessary.
- Similarly, you should be able to get by with a single attack method, so only one of One-Handed, Two-Handed, Archery, and Destruction is really crucial.
- Destruction is the only one that provides any real benefit to a player focused on the others; a few perk points in the Destruction perks that increase specific elements' damage can help increase the power of damage-enchanted weapons.
- Although it's mostly there to reduce damage, Block is often still quite useful when paired up with a strong armour skill, but obviously won't be of much use paired with Two-Handed or Archery.
The other magic schools besides Destruction also often have some form of combat capability, but tend to do more, so they aren't always going to be completely wasteful:
- If you already have armour on, the Alteration armour spells aren't going to give you much, and therefore the perks for those spells won't help much, but Candlelight, Magelight, Waterbreathing, Telekinesis, Detect Whatever, and Paralyze all still make the school very much worthwhile to invest in.
- Restoration can be left mostly perkless if your armour is strong enough, or you have good Alchemy, but the ability to restore stamina with Respite, and the ability to heal followers, can still make the school very compelling to a player who never needs much healing.
- Conjuration has three main components. The actual summoning of creatures probably won't be worth the perks if you're trying to save them. Bound weapons, however, are incredibly useful, especially early on, and especially to an archer (Bound Bow is really, really powerful), and might let you drop Smithing/Enchanting, which are normally perk-hogs, except that you may still need those for armour. And, of course, no matter what, the Novice/Apprentice cost-reducing perks are very much worth the points to most players who want to level enchanting, since you'll need to cast Soul Trap quite a bit.
- Illusion's spells mostly focus on crowd control, or stealth. With enough mana to cast Muffle, or the perk to reduce its cost, you can probably get away without as many perks in Sneak, except that repeatedly casting muffle requires at least four perk points, to get Quiet Casting, and for five perk points in Sneak, you can get Silence, which is basically the equivalent of a permanent muffle effect. And, of course, you can skip out on both Muffle/Invisibility and Sneak if you instead just charge into situations without being sneaky at all, only sneaking as needed to level the skill.
Ultimately, in practice, skipping entire skills isn't necessarily the way to go; instead, you will need to look at things like "How am I going to deal damage?", "How am I going to stay alive?", etc., and focus on perks that help the way you chose. But you will also need to remember that, if you plan on primarily putting perks into Heavy Armor over Light armor, you will at some point be using Light Armor anyway if you plan on leveling all skills to 100.
All of that having been said, there is one major flaw in the whole plan; level 81 isn't the level cap any more. More accurately, there is no level cap. Since version 1.9, any skill that reaches level 100 can be made Legendary, which resets the skill's level to 15, and refunds all perks. When doing this, you keep all character levels you gained from it, so you can keep Legendarying skills over and over to reach far higher levels. But, this doesn't make reaching level 81 any less worthy of bragging rights; mostly, it just gives you more options for perks.
For example, you can get your Smithing, Enchanting, and Alchemy skills to 100, produce a few dozen different sets of armour, each top-quality and enchanted for a variety of builds (so you can swap as you switch which skills you're leveling). Then make hundreds upon hundreds of powerful potions, and reset all three. You will get back all the perk points, and be able to re-use them elsewhere.
Another thing that can help immensely in trying to level like this, is the ability gained at the end of the Dragonborn DLC to reset the perks in a skill. It costs one Dragon Soul to perform, but it leaves the skill's level as it is. The advantage here is, you could level, say, Archery to 100, then decide to level One-Handed next. If you Legendary your Archery skill to get the perk points back, then you're left at level 15 Archery. If instead you use the Dragonborn reset, you get all your perk points back, but still have good Archery skill. That way, if you get into trouble, you can still switch back to your bow for some extra damage. The same idea applies to armour types; switching can be painful, but if you keep your old armour around, and keep the old skill leveled, you can swap back quickly.