Many people believe that Mcclellan had the opportunity to end the civil war at this battle.(1862 instead of 1865) He had enough men in reserves located nearby that he never used. Additionally, he gave Lee's forces ample time to retreat after the end of the battle. He even refused to pursue Lee's army for about a month which caused Lincoln to remove him from his position.
I'm more interested in Mcclellan's motivations but other information will be enjoyed.
General McClellan, in short, was very full of himself and was very much convinced that he was a far better leader than Abe Lincoln could be. For instance, in one letter home to his wife, dated July 27, 1861, McClellan wrote "...by some strange operation of magic I seem to have become the power of the land. I almost think that were I to win some small success now I could become Dictator or anything else that might please me..." In another letter to his wife from October 11, 1861, McClellan decided to insult the President and his cabinet, even going so far as calling Lincoln "...nothing more than a well meaning baboon." Clearly, McClellan believed himself far more important and irreplaceable than he really was.
Another problem the McClellan had was his extreme caution. McClellan had a tendency to grossly overestimate the number of troops he was up against, as would hesitate as a result. This hesitation would prevent his Peninsula Campaign from succeeding in capturing Richmond, and eventually would lead some issues with Antietam.
Finally, another note about him, which ties in with the "full of himself" part is that he would not take blame for defeat. For instance, in his defeat in his Peninsula Campaign, McClellan asserted he could not be to blame for the defeat for he did not have enough men (he outnumbered the Confederates). He went so far in a telegraph (dated June 28, 1862) to Edward Stanton, Secretary of War, to lash out at both Stanton and Lincoln saying "If I save this Army now, I tell you plainly that I owe no thanks to you or any other persons in Washington. You have done your best to sacrifice this army."
After the disastrous Peninsula Campaign, Lincoln had already lost faith in McClellan, and with the events of Antietam, McClellan was removed and immediately replaced by Ambrose Burnside.
I do hope this helped, and if anyone else knows better than me on this, please correct me!
At the battle there is nothing more to be done. McClellan was incredibly aggressive and was perhaps let down by Burnside.
After the battle McClellan does pursue. Hence the actions of 5th Corps at Shepherdstown 19th-20th September, and 6th Corps at Williamsport. McClellan sent 12th Corps to reoccupy Harper's Ferry and his intent was to immediately shift his remaining corps (1st, 2nd and 9th) into the Shenandoah Valley and cut Lee off from Richmond.
Halleck forbid such a movement (and Lincoln hated it), and since McClellan's bridging train was still at Washington was able to prevent McClellan's aggressive movement by refusing to send forward the train. The bridging train was still being held back at Washington several months later, and it's tardy movement to Burnside was implications for Fredericksburg.