Functional difference of Benedict's solution and Fehling's solution
Fehling's solution consists of two parts, Fehling's A and Fehling's B. Fehling's A consists of aqueous copper(II) sulfate, while Fehling's B consists of aqueous sodium potassium tartrate ($\ce{NaK(C4H4O6)}$). Benedict's reagent consists of sodium carbonate, sodium citrate ($\ce{Na2(C6H6O7)}$), and copper(II) sulfate dissolved in water. It's pretty clear now, that the main difference between the two reagents is the complexing agent (citrate in Benedict's and tartrate in Fehling's).

Fehling's reagent was invented by Hermann von Fehling in 1849. You notice that this solution is given as two separate solutions, and is only mixed together while performing a test. This is because the complex formed when mixed ($\ce{[Cu(C4O6)2]}$) is kinetically favored, and that $\ce{Cu(OH)2}$ will eventually precipitate out due to being favored energetically. That's why the two solutions are stored, and only mixed during testing.


On the other hand, Benedict's reagent was discovered by Stanley Rossiter Benedict somewhere in between 1904 to 1936. Thus Benedict's reagent was developed after Fehling's was. The citrate complex in Benedict's reagent is much more stable that $\ce{Cu(OH)2}$, so Benedict's reagent doesn't precipitate on standing for long, unlike Fehling's. This is quite advantageous since you wouldn't have to mix two solutions together to perform the test. This is where Benedict's reagent is quite handy.



Both the reagents have a similar way of oxidizing aldehydes. The copper complex oxidized the aldehyde to a carboxylic acid, and while reducing itself from $\ce{Cu^2+}$ to $\ce{Cu+}$ 
 ($E^\circ = \pu{0.154 V}$). This reduction potential is sufficient to oxidize aldehydes to carboxylic acids. There is not much difference here though.

Neither of the reagents can oxidize aromatic aldehydes, due to the fact that they lack α-hydrogens, and thus oxidizing them requires stronger oxidizers (like Tollens’ reagent).

In short, Benedict's reagent is kind of like an "improved" version of Fehling's reagent, which is much more stable due to the difference in the complexing agent.