Disregarding lore, UI, character design, character count, and overall polish (where blizzard/overwatch clearly shines), there are tons of differences in gameplay between the two games. I'm going to list some of the main ones that I have thought of, hopefully people will read this and contribute to it so it can be even more comprehensive.

5v5 locked heroes vs 6v6 unlocked heroes

• This means champions in Paladins are inherently stronger than their counterpart in Overwatch
• Each player influences their team and the outcome of the game more
• Paladins: Can't Swap heroes, can't have Duplicate heroes. More focus on team composition, and drafting picks (ranked).
• Overwatch: Can Swap heroes, can have Duplicate heroes. More fun, more casual, and there is also some strategy in swaps!

Loadouts! (Paladins Only)

• Paladins lets you change your build outside of the game that may change your characters playstyle drastically. This brings "netdecking" to Paladins and lets players theorize multiple loadouts to tinker with to have for future games. Overall: this gives more strategical depth to Paladins
• Overwatch: No character building

Item buys (Paladins Only)

• This one is less important/creative/interesting. There are smaller purchasable stat changes in-game that let you adapt to the opponents team composition. Stuff like: % Damage to Shields, Reduced Healing on hit, Cooldown reduction, Mount speed, CC reduction.

Siege Game Mode vs Hybrid(Assault Escort)

• Siege is the main game mode in Paladins casual/ranked.
• Assuming you know how Overwatch's Hybrid works (capture point, then escort payload to the end), Siege is somewhat similar, yet plays out quite different. There is no team swapping. Maps are completely symmetrical, and the point is always in the center. Games are played first to 4/4, you can earn a point by: taking point, pushing payload after, or defending payload after (except match point eg: 3/4). There are no TIES.
• This means in Siege mode the teams are grouped more often because teams respawn together after each round, similar to Overwatch's Control/KoTH mode. So basically OW's Hybrid mode, but more fair!

Symmetrical maps vs Asymmetrical maps

• Disregarding Paladins' extra Payload game mode, all maps are symmetrical, whereas Overwatch most maps are asymmetrical escort/assault/hybrid with teams taking turns attacking/defending.
• Asymmetrical maps in Overwatch gives hero swapping strategy. It tests your map knowledge more. Some heros are better on certain parts of the map than others.
• Symmetrical maps in Paladins gives team composition and character choosing more importance. Maps are easier to learn. There are no map imbalances for sides.

TTK and Strength of Ultimates

• The TTK (time to kill) is higher in Paladins, meaning characters die slower, live longer, and don't get one-shot often.
• Overwatch: ultimates are more potent and stronger. The TTK for some ultimates are instant, meaning there are lots of ults that one-shot you.
• Paladins: ultimates are weaker. There's only one(Drogoz) that can one-shot a singular hero, and it can be juked or shut-down. There are no resurrection ultimates, only Fernando that keeps team members alive for a few seconds. Most of the ultimates don't activate instantly, allowing for outplays. This means games are less decided on charging ults and winning games with them, and more on autoattacks, skill shots, and overall more interactions between players. Strongest and Fastest ult is probably Androxus.

Ranked

• Paladins: there's only 1 game mode (Siege). Only Solo/Duo Queue, no premades. There are no ties. There's draft pick, and your heros are locked. Team composition is urgent during this phase.
• Overwatch: there are 3 game modes (Escort, Assault, Hybrid, Control). More variation and lots of ties. Can change heros.
• I know most players don't like playing some game modes, and wishing they could ban certain ones. I personally think Paladins Siege is the best way to play, it incoporates the best parts of Escort/Assault and there are no ties. I also think having no premades in Ranked is important for fairness, many people complain about being teamed vs premades in Paladins casuals and in Overwatch.

Mounts (Paladins Only)

• In Paladins you gain mounts every time you spawn. You can get dismounted by being attacked, or manually clicking. You cannot remount.
• This allows for larger maps, making the main control point farther from spawn. Spawns are always static, but you can affect the time it takes to get back to battle by buying mount speed.
• Dismounting players is a strong flank tactic that severely cripples and delays teams from getting back to battle.
• Flanks can run more obscure paths with their mounts to position themselves faster. Teams can also travel as 5 and get behind the opponents to do a coordinated flank much more quickly with mounts.
• As you can see the mounts bring more strategy and depth to Paladins

Health Pickups (Overwatch Only)

• Overwatch: Yes there are health pickups.
• Paladins: Not anymore. You can heal by sitting out of combat for a certain time (there's a UI for that) or from supports.
• I think this is an important distinction: Poking is more effective in Paladins, taking damage is more punishing (especially with no supports), and regenerating health to get back to the fight isn't as simple.

Shields

• Overwatch: Reinhardt (Frontal), Winston (Bubble), Zarya (Singular), D.VA (Singular), Lucio? (Bonus HP ult)
• Paladins: Fernando (Frontal), Makoa (Bubble), Barik (Frontal), Ruckus (Singular Damage Reduction)
• Reinhardt vs Fernando: Reinhardt has no cooldown (unless broken), and instead regenerates his shield hp. Fernando has a cooldown and a time limit, can't activate his shield whenever, and generates a new shield every use. It's hard to compare their values (2k vs 8-10k) since damage/hp values are different. Reinhardt has the stronger shield since he can hold it up forever and recharges it in 3 seconds when broken, wheras Fernando has 10 seconds to hold his shield, and recharges in 8-15 seconds depending on damage taken.
• Shield Popping: In Paladins you can pop shields by disabling/ccing the tank. Fernando you can get behind him, Makoa you have to be inside his bubble. Once disabled they can't reactivate the shield until the cooldown is over. This is a strong, advanced tactic that adds depth and counterplay to shields in Paladins.
• Wrecker: In Paladins you can buy an in-game item called wrecker to deal increased damage to shields. Usually bought on DPS characters to melt shields faster and to counter 2-4 tank compositions.
• The 3 above points show that shielding in Paladins has more counterplay to it and stresses more careful usage/timing of shields.
• Paladins also has an additional team shielding character (Barik) compared to Overwatch. Makoa's shield is also more potent than Winston's, and has a stronger kit overall.

Match History/Stats

• Overwatch: No Match History, No shown stats of your teammates.
• Paladins: Buggy Match History (Paladins.guru/Masterpaladins fixes that and improves on it), shows stats of both teams in after-game lobby.
• I could see hiding stats in Overwatch as a design choice to avoid toxicity, but not having match history in 2016 is pretty bad. Stats in general are way better in Paladins

(UI) HP Bars and Damage Numbers

• Overwatch: Enemy HP bars only show when you hit them, no damage-numbers in game.
• Paladins: Enemy HP bars ALWAYS show when they are in your line of sight. Damage numbers show and you can adjust the size of them.
• This means Paladins has a more moba-like feel, since in mobas it's important to know hp values and damage values, while Overwatch keeps a more shooter-like feel where you can't always see their hp (csgo).
• Therefore Overwatch requires more communication, optimally voice, for a team to succeed. This means quick play in OW will inherently be harder to secure kills and have teamwork moments happen if noone communicates.
• In Paladins, you have more in-game voice commands through the V key. This makes it easier to communicate to your team without a mic. In casual games (aka quickplay), it will also be easier to work together since everyone can see everyone's hp, so people know who to focus first.

Voice Chat

• Overwatch: Yes
• Paladins: Not yet (doesn't seem immediately planned either)
• Communication will be easier on Overwatch, while it is also more important.

Less important:

• Obviously champions are different, and exploring new champions is always fun! There are different skill sets and plenty unique skills, character concepts, and ideas. Sure some might seem "stolen" from Overwatch, but keep in mind a lot of Overwatch character designs aren't completely original in the first place either! And also Mei stole Evie's ice block concept, not the other way around xd
• Payload in Paladins has no overtime. You either push farther than them in a certain time-slot, or you lose. If you pushed the payload 100% in 3 minutes, than they only have 3 minutes to do the same. I think this is an improvement compared to Overwatch's Escort game mode.

I just wanted to do this write-up so future players can clearly see the differences and maybe stop disregarding Paladins as a "clone" and have a try themselves (it is free2play after all). I might be missing something, so I'm up for suggestions to add to this thread! Thanks for reading. (This was made during OB35 patch)

update: expanded on TTK and added Shields section

update 161119(date): played some more ow cuz free weekend, noticed some more differences: Match History/Stats, HP Bars, etc