Sanders played from 1989 to 1998. Emmitt played from 1990 to 2004. The fact that they were both among the best ever RBs and started their careers one year apart is why they are always compared.
1st 10 seasons
So, first let's look at how Emmitt's first 10 seasons compared to Barry's 10 seasons. Regular season statistics:
Barry (1989-1998): 153 games; 3062/15269/99 rushing (4.99 ypc, 99.8 ypg); 352/2921/10 receiving (8.3 ypr); 18190 YFS; 109 total TDs; 41 fumbles
Emmitt (1990-1999): 155 games; 3243/13963/136 rushing (4.31 ypc, 90.1 ypg); 442/2728/11 receiving (6.2 ypr); 16691 YFS; 147 total TDs; 45 fumbles
Rushing yardage
Sanders was clearly a better runner. He was much better in ypc and consequently ended up with 1306 more rushing yards in 2 fewer games and 181 fewer carries... that is a very impressive advantage for Sanders. And it doesn't even factor in the fact that Smith played on a better team with better offensive teammates and better defenses... the better offenses presumably made it easier for Smith to perform well, and the better defenses were likely largely responsible for the extra carries Emmitt got in comparison to Sanders... without those extra carries, the results in this category would have been even more lopsided.
Big edge to Sanders.
Rushing TDs
Clearly Smith had a huge edge in rushing TDs. Some of the difference could be explained by the advantages cited above (i.e., better teammates), but not a disparity of 37 rushing TDs. Clearly, Smith had a nose for the end zone and was an excellent goal line back. As we know, Sanders was pulled at the goal line a fair amount in his career.
Big edge to Smith.
Receiving
Smith had 90 more receptions, yet Sanders had more receiving yards and almost as many TDs. Once again, Sanders was more productive with his opportunities, again underlining that Sanders is a better runner. Given that Emmitt had the 90 additional receptions, and given I don't know anything about their catch percentages, targets, drops, etc., I'm not sure it's fair to say Sanders was a better receiver, though it certainly seems that way... regardless, we know he did more with the ball once he caught it.
Small edge to Sanders.
Blocking
Not sure how to measure this objectively, but I recall Smith being known as a good pass blocker and Sanders being viewed as average to below average. However, I don't think this was as big a factor as other things.
Small edge to Smith.
Durability/Toughness
I think Emmitt is known for his toughness. I don't recall it being viewed as strongly as a positive for Sanders as for Smith, but the fact remains that Sanders played in 153 games, compared to Smith's 155.
Wash.
** Update **
Up to this point, I'd give Sanders a slight edge, since I think rushing yardage trumps the other categories, but it's very close thanks to Emmitt's huge edge in TDs.
Situations
Emmitt played with the following Pro Bowl offensive teammates: Troy Aikman (6), Michael Irvin (5), Jay Novacek (5), Daryl Johnston (2), Nate Newton (6), Mark Stepnoski (3), Erik Williams (4), Mark Tuinei (2), Ray Donaldson (2), Larry Allen (5). That's 40 Pro Bowl teammates in 10 seasons, including 24 run blockers (OL & FB). Plus, Emmitt played his first 5 seasons for Jimmy Johnson, a great coach. Emmitt's teams also finished in the top 10 in yards allowed in 8 of those 10 seasons and in the top 10 in points allowed in 7 of those 10 seasons, which presumably gave him more opportunities.
Meanwhile, Sanders played with the following Pro Bowl offensive teammates: Lomas Brown (6), Herman Moore (4), Kevin Glover (3). That's 13 Pro Bowl teammates in 10 seasons, including 9 run blockers. Sanders never played for a coach as good as Jimmy Johnson. His teams finished in the top 10 in yards allowed and points allowed just once each in his 10 seasons.
This is a huge disparity. We could get into comparisons of All Pro teammates and HOF teammates, and they would show a similar disparity.
This factor is often stated as Emmitt having a better OL, but, in reality, Emmitt had better coaching, better OL, better passing game, better defense... heck, without looking into it, I wouldn't be surprised if his teams had better special teams, too.
Big edge to Sanders.
Honors/Awards
This category reflects how they were viewed in comparison to their peers. Obviously, this is important to this comparison, since they were peers to each other.
Emmitt:
1990 NFL AP Offensive Rookie of the Year
1992 NFL Newspaper Ent. Assoc. MVP
1993 NFL AP MVP
1993 NFL PFWA MVP
1993 NFL Newspaper Ent. Assoc. MVP
1993 NFL Bert Bell Award (Player of the Year)
1993 NFL Super Bowl MVP
4 1st team AP All Pro selections
1 2nd team AP All Pro selection
8 Pro Bowl selections
Barry:
1989 NFL AP Offensive Rookie of the Year
1991 NFL Bert Bell Award (Player of the Year)
1994 NFL AP Offensive Player of the Year
1997 NFL AP MVP
1997 NFL PFWA MVP
1997 NFL Newspaper Ent. Assoc. MVP
1997 NFL Bert Bell Award (Player of the Year)
1997 NFL AP Offensive Player of the Year
6 1st team AP All Pro selections
4 2nd team AP All Pro selections
10 Pro Bowl selections
It looks close, with a small edge to Sanders until you reach the All Pro selections, where Sanders doubled up Smith, which is extremely impressive. Sanders was no worse than 2nd team All Pro in every single season of his career... I wonder if any other player can say that. Obviously, voters thought Sanders was significantly better.
Big edge to Sanders.
Emmitt's extra 5 seasons
Emmitt holds the records he has because he played 5 more years than Sanders. There is no reason to believe Sanders wouldn't have continued to outperform Smith into their declining years, particularly given how strong their respective 10th seasons were... but the fact is that Emmitt played on and Sanders didn't. Here are the numbers in those 5 seasons:
71 games; 1166/4392/28 rushing (3.77 ypc, 61.9 ypg); 73/496/0 receiving (6.8 ypr); 4888 YFS; 28 total TDs; 16 fumbles
On the one hand, there is no doubt Emmitt provided value to his teams over this period, or he wouldn't have gotten those 1200+ additional touches. So there must be some positive value in that. However, his effectiveness dropped off sharply, and he did not make the Pro Bowl or earn any other notable honors during those 5 years, so the value is limited.
Small extra credit edge to Emmitt.
** Update **
At this point, IMO Sanders has a huge lead over Smith.
Postseason
There really is no comparison here:
Emmitt: 17 games; 349/1586/19 rushing (4.54 ypc, 93.3 ypg); 46/342/2 receiving (7.4 ypr); 1928 YFS; 21 total TDs; 3 championships; 1 Super Bowl MVP
Barry: 6 games; 91/386/1 rushing (4.24 ypc, 64.3 ypg); 21/111/0 receiving (5.3 ypr); 497 YFS; 1 total TD; 0 championships
Though it isn't Barry's fault he didn't have more opportunities, the fact is that his performance regressed in the postseason, while Emmitt's improved.
Big edge to Smith.
Conclusion
All things considered, it is close. But I don't see how Emmitt's postseason accomplishments make up for the substantial edge Sanders had when everything else was considered. Especially considering that the supporting cast issue applies to postseason just as much as it does to regular season (to state the obvious).
So: Barry > Emmitt