Of the 361 spells present in the 5e Player's Handbook, 62 are new and not present in 3.5.
I didn't count spell renames (Minor Image-> Minor Illusion) or spell merging/splitting (such as the Cure Wounds line) as different spells.
These are the new spells:
- Arcane Gate
- Armor of Agathys
- Arms of Hadar
- Aura of Life
- Aura of Purity
- Aura of Vitality
- Banishing Smite
- Beacon of Hope
- Beast Sense
- Blade Ward
- Blinding Smite
- Branding Smite
- Chromatic Orb
- Circle of Power
- Compelled Duel
- Compulsion
- Conjure Barrage
- Conjure Volley
- Cordon of Arrows
- Crown of Madness
- Crusader's Mantle
- Destructive Smite
- Dissonant Whispers
- Druidcraft
- Elemental Weapon
- Enhance Ability
- Ensnaring Strike
- Feign Death
- Find Familiar
- Fire Bolt
- Friends
- Giant Insect
- Grasping Vines
- Guardians of Faith
- Guiding Bolt
- Hail of Thorns
- Healing Word
- Hellish Rebuke
- Hex
- Hunger of Hadar
- Hunter's Mark
- Lightning Arrow
- Mass Healing Ward
- Misty Step
- Moonbeam
- Phantasmal Force
- Poison Spray
- Power Word Heal
- Prayer of Healing
- Sacred Flame
- Searing Flame
- Spare the Dying
- Staggering Smite
- Swift Quiver
- Thaumaturgy
- Thorn Whip
- Thunderous Smite
- Thunderwave
- Vicious Mockery
- Warding Bond
- Witch Bolt
- Wrathful Smite
For completeness, the renamed spells that I didn't consider as new are:
- antipathy and sympathy to antipathy/sympathy
- clairaudience/clairvoyance to clairvoyance
- summon nature's ally to various conjure X spells
- create water to create or destroy water
- major creation to creation
- cure light wounds et al. to cure wounds
- detect evil and detect good to detect evil and good
- detect poison to detect poison and disease
- dispel evil and dispel good to dispel evil and good
- holy word to divine word
- phantom steed to find steed
- inflict light wounds et al. to inflict wounds
- detect animals or plants to locate animals or plants
- magic circle against alignment to magic circle
- minor image to minor illusion
- programmed image to programmed illusion
- protection from evil and protection from good to protection from evil and good
- neutralize poison to protection from poison
- ray of sickening to ray of sickness
- telepathic bond to telepathy