While I was playing the game I realized that I needed a back-up weapon to conserve my bullets. I need to know what are the advantages and disadvantages of these two skills. So people can you help me?

I'd personally go unarmed. Later on you can get knockdown techniques, paralyzing techniques, different blows you can deal, dispersion and ballistic gloves which work wonders, even the spiked knuckles are pretty decent. At bare minimum it will give you a few seconds to finish someone off with a bullet to the head.

Also with bare knuckles you don't have to worry about weapon condition, plus there is a perk later on which reduces/bypasses a set amount of DT which while you need a high unarmed to get also affects melee weapons as well.

Though my game isn't shipped yet, judging from previous responses, unarmed is superior [from my point of view], dew to avilability of special techniques, and perks, that would allow, like in all previous Fallout game [save original, due to rather low level cap] a playstyle of unarmed and unarmoured demigod of the wastes, relying on non-combat skills and stealth, when not delivering special criticals. [Shame Slayer was nerfed.]

So what about melee?

Melee will give you greater variety of tools of trade, along with possibility of poisoning 'em, and greater range, at cost of having an item with degrading condition.
Again, there may be some other advantages and disadvantages, but the greatest advantage of unarmed, in my opinion, is being able to successfully defend yourself - or assault anyone unexpected, without having any visible/stealable/otherwise unreliable weapon.

I never used Melee in FO3 what with the massive haul of weapons, but in NV I find that the scarcity and the DT neglifying a lot of basic guns, Melee comes in handy. I have a Melee of 20 SP yet I have used about 50% of my combat with Power Fist - an excellent weapon I must say. I feel rather saddened that I did not use it in FO3.

It's odd since I hated Melee in FO3 for some reason but the other day I got the Achievement/Trophy for X Damage by Guns and X Damage by Melee within 10 minutes of each other.

EDIT: DenSlaver21688: Thank you for pointing out the massive hole in my logic, it is in fact Unarmed. Apologies.

Reno

Power fist is an unarmed weapon, melee is classified as non-explosive thrown, blunt, or bladed (excluding the bladed fist or whatever its called) i believe.

...afterthought, this game made me miss the mega power fist, but I think if they actually added it you could probably take out a deathclaw in one hit.

ballistic glove + piercing Strike= bad ass

umm

Ninja, Extra hit perks and high sneak and you take Deathclaw/super mutants who have DT 30 on very hard and hardcoe with 1 hit with supersledge [especially the Oh, baby!.

Also melee weapons also have special attacks and if you like fighting outside of vats take slayer perk... youll swing so fast iyoull kill muties in 2-3 hits, also power attacks are done quicker and these ignore A LOT of enemy DT.

I used sledge hammer a lot even as revolver/sniper guy but i had ninja perk for better sneak attacks [also had professional, cowboy and handloader for insane The Lucky shots]

And even as gunner melee weapons were great. Ballistic fist is cool but i dislike its animation kinda lol. and that it knocks enemies back so i gotta run to them again