From Diablo Wiki
Diablo III's maximum character level at launch was 60, which players were meant to reach by the end of Hell difficulty. This was a big change from Diablo 2's system of long term grinding to the maximum level of 99, but the Diablo 3 developers didn't think that system was ideal for Diablo 3.
They changed their minds not long after release, and while the maximum level remained at 60, the Paragon system was added to give players an additional 1-100 levels to grind once they reached level 60. (Each Paragon level rewards the usual stat bonus, plus a permanent passive +3% bonus to MF/GF.
The maximum level of 60 was a fairly late change during development, with early comments from the devs pointing to level 99 or 100 for the maximum. After initial skepticism and disagreement, most fans accepted the logic behind the change, and a site vote on Diii.net found nearly 50% of fans strongly supportive of the change, with just 16% objecting to it. That early support quickly eroded as fast players got to the maximum level and found Inferno play very difficult, as well as insufficiently-rewarding without any more levels to gain or other tangible signs of their long term progress.
One of the major player complaints/comments about the change to 60 was that it was purely made to allow expansion packs to raise the maximum level. That still may happen, but with the Paragon level system it's unclear how maximum levels will be raised, if at all, in an expansion pack. If each class gets more skills or skill runes, would the maximum level be extended to 70 to allow those new skills to be spread out? But when what of characters already well into the Paragon levels; would their experience suddenly start counting towards level 61-70, before reverting to Paragon levels once they maxed out again at level 70? Perhaps it would be easier to raise the Paragon max level to 120 or 150, purely to give players more leveling up to do in the expansion.
The Level 60 Debate
The level 60 decision was reached in 2010 when many game features were far from their final state. At that point skill points and trait points were still in the game, so much of the debate over the issue revolved around how powerful characters would feel with fewer such points to allocate to their active and passive skills. The other main complaint was that the end game would be boring without any sort of ongoing character progression.
The developers discounted those concerns and insisted that the end game would offer plenty of fun even without any sort of character building. Ultimately, their predictions on that front proved incorrect, and most fans post-release were displeased with the lack of end game activities and no sense of character progression. This realization prompted the introduction of the Paragon Level system in v1.0.4, in August 2012.
The Paragon Levels System
Shortly after Diablo III's launch, many fans found the end game lacking. The difficulty balance of Inferno levels was much debated and constantly tweaked by Blizzard in patches, but an additional area of discord was the lack of a sense of character progression. With the max level achieved by a single play-through of the Normal/Nightmare/Hell content, the end game became entirely about grinding for items, and with Diablo III's item system built largely around the Auction House, most players very seldom found any item upgrades. Players felt they were simply grinding for gold, which they could then spend in the Auction House to obtain better gear.
Blizzard attempted to address this complaint with numerous re-balances to Inferno, including much increased item drop rates, but they also implemented the Paragon Levels system, which adds 100 more levels (achieved from experience gain) once a character achieves level 60. With each Paragon Level a character gains the same attribute bonuses as a normal level up (3 to main stat, 2 vitality, 1 each to the non-main stats), plus 3% better Magic Find and Gold Find. This feature works hand in hand with a new Magic/Gold Find cap of 300%, so that a character at the maximum Paragon Level 100 would have maximum gold and magic find without using a single item to boost those stats.
How well this new feature will address fan complaints about item finding and end game enjoyment remains to be seen, but the feature seems popular in the days after v1.04's debut.
Original Arguments for the Level 60 Change
Blizzard argued for the level 60 change (from their earlier projections of a max level of 99 or 100) on several topics. The first was skills, largely in regards to how skill points would work. This entire argument ultimately became irrelevant with the removal of skill points, but when it was made in 2010 it went much like this:
With just 60 skill points the game could be balanced to make each Skill point feel valuable. With more skill points they must be balanced to do less each, or else characters would become overpowered.  Plus the higher level skill points will still be and feel valuable in D3, rather than just adding .2% to some passive skill, or to a 3rd or 4th synergy skill, as they did in D2.
- The same principle applies to trait points, naturally.
End game fun. The developers promised that there would be better things to do in Diablo III's end game than endlessly grinding the same level or boss.
Characters do not feel "finished" when they reach Clvl 60. In D2, players tended not to play characters anymore once they'd reached the highest level. Most players were bored of that char by then, after all the grinding, and it seemed pointless to play when you were not gaining experience. This is not at all the case in WoW, where characters reach the maximum level before finishing all the content, and are then still played to raid the hardest dungeons, looking for top items. In WoW (and some other RPGs) characters essentially *have* to be max level to handle the hardest dungeons. This might be what the D3 team is planning; end game dungeons that are only doable for expert players with very powerful characters, who would have fun and item hunt and earn achievements by doing them.
More Levels Just for Fun?
Long prior to the Paragon system, many fans suggested that the experience counter keep rolling, or that levels beyond 60 exist just to give players something to do, even if those extra levels didn't add any additional bonuses. Bashiok said no:
The developers have not denied that they'll add more Clvls in expansion packs, and that lowering the max level from 99 to 60 makes this easier/more natural (going from 60 to 70 or 75 seems more natural than 99 to 110, or whatever) but they have bristled at fan allegations that it's their only, or chief, motivation. 
Character Level Progression
The level progression curve is being designed to characters should reach 30 at the end of normal difficulty, 50 at the end of Nightmare difficulty, and 60 at the end of Hell difficulty. The 30/50/60 progression was made pre-game, and it's roughly accurate for players who clear most of the content, don't skip a lot of areas, and don't play with excessive +experience equipment or on a very high Monster Power setting. Many players chose to entirely subvert that process and get a friend to rush them, then level up by leeching experience as their much higher level friend blasts through the monsters on a high MP level.
Another argument about the level 60 decision was that it felt like too low a number, but in September 2010 Bashiok argued that the number was irrelevant:
Max Without Grinding
Allowing characters to max out their level without endless grinding is one of the main stated reasons for this change. The D3 Team thinks they can offer a much more fun and varied end game than endlessly running the same level or boss, and they do not want to force players to endure that style of play in order to reach the max level. There are numerous reasons for this.
The Arena, for one. The team announced the Clvl 60 change long before they revealed their plans for Arena battles, but once that game feature was known, the connection became obvious. Players will want to duel with their high level characters, and since character power is closely tied to Clvl, the developers wanted to level that playing field. By designing the game so that characters could reach the maximum level through the normal content, players can create top level dueling characters without spending hundreds of repetitious hours. (Freeing them up to spend those hundreds of repetitious hours hunting items to use in the Arena.)
As with most of the rest of their arguments, the Paragon system proved their original logic flawed.
Arguments Against Sixty
Most players agreed with the developers' logic on the September 2010 level 60 announcement, but there was still quite a bit of debate over making the maximum level so easy to reach. In the earlier games in the series, building a maximum level character was a sign of devotion, skill, and a lot of time spent playing. That distinction was gone in Diablo III (before the Paragon system), since anyone could get to Clvl 60 just by completing the base content.
The counter argument by the developers is that Diablo III will have much better things to do in the end game than just grinding the same boss over and over again. They've assured fans that there will be plenty of things to do long term, and that there will be better ways to show off your epeen than a character in the 90s. Those other options haven't been revealed yet, but the team has said there will be high level Achievements, and that characters will fight for ranking on some sort of Arena Ladder. Other end game diversions have yet to be revealed.
Skeptical fans also pointed out that Clvl 60 was the same maximum level cap that World of Warcraft offered, before that was raised by 10 the first two WoW expansions, and 5 more in the third. Since at least two expansions are planned for Diablo III, and since each will add another Act to the game, along with other content, Diablo III following in WoW's footsteps with an increased Clvl in each expansion seems a safe bet.