Why are asymmetric cryptography keys more vulnerable to brute force attack than symmetric ones? I came across this paper which says that 


  Asymmetric keys must be many times longer than keys in
  secret-cryptography in order to boast equivalent security.  Keys in
  asymmetric cryptography are also more vulnerable to brute force
  attacks than in secret-key cryptography. There exist algorithms for
  public-key cryptography that allow attackers to crack private keys
  faster than a brute force method would require.


I would like to know:


What makes Asymmetric Cryptography keys more vulnerable to brute force attacks?
In which cases should Asymmetric Cryptography usage be avoided?
For your first question:
The main point here (at least that comes to mind) is that of how the key is made, used, and subsequently how it is attacked. Good symmetric ciphers are designed so that the best possible attack is brute force, i.e. simply trying with each possible key, which are typically random (or as good as). With a 128-bit key you have $2^{128}$ possible keys. Trying all of these takes a lot of time. 

With an asymmetric cipher however, the private and public keys have a mathematical relation between them, and asymmetric ciphers are based on problems that are computationally hard to solve. An example of this is RSA, where the keys are based on two large (and secret) prime numbers, $p$ and $q$, then multiplied to create an even larger number, $n$. The result is then used for encryption and decryption (i will not go into further details of RSA here). To crack this system the most straight forward way is to find $p$ and $q$ by factoring $n$ (which is not secret).

Checking $2^{128}$ (or $2^{127}$ on average) symmetric keys with todays computational power is simply not possible within any conceivable time-frame. Factoring a 128-bit number however, takes about a second (depending on hardware and optimization). Thus, one needs larger keys for RSA than symmetric ciphers, e.g. 128-bit symmetric keys are typically approximated to equal 2048-bit RSA keys.

For your second question:
Encryption and (even more so) decryption with asymmetric ciphers are often a lot more computationally intensive than a symmetric cipher (if i remember correctly is RSA typically about 1000 times slower than AES). This makes asymmetric ciphers impractical for encrypting large chunks of data. Subsequently, for encrypting e.g. internet traffic: asymmetric ciphers are typically used for securely exchanging keys that are used to encrypt/decrypt using a symmetric cipher.

EDIT: As rightly pointed out by @fgrieu the statement "128 to 256-bit symmetric keys are typically approximated to equal 2048 to 4196-bit asymmetric keys." is not correct, and comes from a writing-mistake on my part. The correct statement was supposed to be that 128-bit symmetric keys are typically approximated to equal 2048-bit RSA keys.