What is the difference between DSA and RSA?

It appears they are both encryption algorithms that require public and private keys. Why would I pick one versus the other to provide encryption in my client server application?

Check AVA's answer below.

My old answer seems wrong

Referring, https://web.archive.org/web/20140212143556/http://courses.cs.tamu.edu:80/pooch/665_spring2008/Australian-sec-2006/less19.html

RSA
RSA encryption and decryption are commutative
hence it may be used directly as a digital signature scheme
given an RSA scheme {(e,R), (d,p,q)}
to sign a message M, compute:
S = M power d (mod R)
to verify a signature, compute:
M = S power e(mod R) = M power e.d(mod R) = M(mod R)

RSA can be used both for encryption and digital signatures, simply by reversing the order in which the exponents are used: the secret exponent (d) to create the signature, the public exponent (e) for anyone to verify the signature. Everything else is identical.

DSA (Digital Signature Algorithm)
DSA is a variant on the ElGamal and Schnorr algorithms. It creates a 320 bit signature, but with 512-1024 bit security again rests on difficulty of computing discrete logarithms has been quite widely accepted.

DSA Key Generation
firstly shared global public key values (p,q,g) are chosen:
choose a large prime p = 2 power L
where L= 512 to 1024 bits and is a multiple of 64
choose q, a 160 bit prime factor of p-1
choose g = h power (p-1)/q
for any h<p-1, h(p-1)/q(mod p)>1
then each user chooses a private key and computes their public key:
choose x<q
compute y = g power x(mod p)

DSA key generation is related to, but somewhat more complex than El Gamal. Mostly because of the use of the secondary 160-bit modulus q used to help speed up calculations and reduce the size of the resulting signature.

DSA Signature Creation and Verification

to sign a message M
generate random signature key k, k<q
compute
r = (g power k(mod p))(mod q)
s = k-1.SHA(M)+ x.r (mod q)
send signature (r,s) with message

to verify a signature, compute:
w = s-1(mod q)
u1= (SHA(M).w)(mod q)
u2= r.w(mod q)
v = (g power u1.y power u2(mod p))(mod q)
if v=r then the signature is verified

Signature creation is again similar to ElGamal with the use of a per message temporary signature key k, but doing calc first mod p, then mod q to reduce the size of the result. Note that the use of the hash function SHA is explicit here. Verification also consists of comparing two computations, again being a bit more complex than, but related to El Gamal.
Note that nearly all the calculations are mod q, and hence are much faster.
But, In contrast to RSA, DSA can be used only for digital signatures

DSA Security
The presence of a subliminal channel exists in many schemes (any that need a random number to be chosen), not just DSA. It emphasises the need for "system security", not just a good algorithm.