How exactly are socionics and MBTI different? Socionics superficially looks exactly as MBTI. 8 functions, 16 personality types. There's even a table, perfectly corresponding socionics types with MBTI types. Then what makes socionics and MBTI importantly differ?
Both socionics and MBTI are a try to dig deeper on the path of Carl Gustav Jung, who observed empirically (both from his work as a psychiatrist and from his travels across the globe) that all human brains displayed common cognitive descriptions, which Jung described by two energy-attitudes, two perception functions and two judgment functions.  


  About Jung (about 100 years ago - his milestone book about cognitive types
  was published in 1921)
  
  The energy attitudes are introverted - the energy flows outward from the
  inside stimulation; and extraverted - the energy flows inward from the
  outside stimulation.
  
  The perception functions are intuition - a misnomer, with intuition you
  take information from a 3,000 ft view and make connections from this - and
  sensation - you take information from your senses and like to get into
  details, like with a magnifier glass.
  
  The judging functions are thinking - you make your decisions based on
  logic - and feeling - a slight misnomer, you make your decisions based on
  your subjective set of values (N.B. objective and subjective are commonly-used
  words in C G Jung's books).  
  
  Following Jung, we all have a piece of everything in us, but we also have a
  preference for each dimension - energy management, taking information,
  making a decision.  
  
  Jung described a set of 2x2+2x2=8 cognitive functions: introverted thinking,
  introverted feeling, extraverted thinking, extraverted feeling, introverted
  intuition, introverted sensation, extraverted intuition and extraverted
  sensation - also known as Ti, Fi, Te, Fe, Ni, Si, Ne and Se. Yes, N stands
  for iNtuition, as i stands for Introverted.  


As mentioned in the comments, the MBTI was developed in the US - Katharine Briggs met Jung in Switzerland and her daughter Isabel Myers-Briggs was raised in a family/friends/colleagues environment that talked about cognitive types and she followed her own path, about 35-to-50 years after Jung.


  About Briggs and Myers (about 50 years ago)  
  
  They described the structure of the preferences of the cognitive functions:
  sixteen combinations rephrase Jung's preferences, with the three axis from
  Jung (energy orientation of the dominant function, perception function,
  judgment function) and with a fourth axis, describing whether the
  extraverted function among the main two functions used by a profile
  is either a perception function or a judgment function.  
  
  This descriptions fits into four letters (E or I, S or N, T or F, J or P)
  that describes the preference of a cognitive profile for two Jungian functions.  
  
  Every profile has an inward function and an outward function, every profile
  has a perception function and a judgment function. With a main (dominant) and
  an auxiliary (auxiliary) function, this makes 16 combinations of letters,
  and 16 archetypes to describe common traits of a profile.  
  
  The Myers-Briggs Type Indicator (or MBTI) is a questionnaire that tries to
  figure out whether your preference is E or I, S or N, T or F, J or P. Its
  main weakness is that this way of typing is not accurate at all - though it
  had a huge success in business. This was a weakness from the science viewpoint,
  as the MBTI never improved upon itself, by trying to describe with better
  accuracy the nature of our cognitive functions (which was Jung's goal).  


Now, we turn to the former USSR, where other people followed the path of Jung,
too. They founded socionics, which is an extrapolation of Jung's work to describe
how the 16 profiles communicate and interact with one another.  


  About socionics (about 50 years ago - USSR)  
  
  Socionics also developed a fourth axis beyond Jung's three axis, but do not
  be misunderstood about it, as it is not the same fourth axis as in MBTI.
  In socionics, the fourth letter describes whether the dominant function of
  a profile is a perception function (p) or a judgment function (j).  
  
  However, while MBTI focused on typing with four letters, socionics kept
  working both on the four-letter code and on the description of the Jungian
  functions. For the four-letter code, socionics described one-letter
  archetypes like in MBTI, but also two-letter and three-letter archetypes.  
  
  About the Jungian functions, socionics described how a given Jungian function
  (e.g. Introverted Thinking) could be described by a slightly different archetype
  depending on its rank within the eight functions - while MBTI described
  mostly the first two or the first four functions of a profile, socionics
  tried to describe the eight functions of a profile, by decreasing order of preference.  


Trying to describe a cognitive profile by an order of functions leads us to
figuring out how stress states may alter our use of our preferred functions.  

This topic was mentioned by Jung (through the concept of Shadow), it is hardly
described in MBTI (not much developed either is the topic of mastering our
functions as we grow older - there are a few life stages when we care
differently for our preferred cognitive functions), while socionics described
a stress model called Model-A (this is a first step on our journey).  


  Wrapping things up - until you ask for more?  
  
  When MBTI and Socionics speak (about) the same with a different vocabulary:
  MBTI  -  Socionics  -  Jungian functions (descending order)
  INTP  -  INTj       -  Ti Ne Si Fe Te Ni Se Fi
  ENTP  -  ENTp       -  Ne Ti Fe Si Ni Te Fi Se
  INFP  -  INFj       -  Fi Ne Si Te Fe Ni Se Ti
  ENFP  -  ENFp       -  Ne Fi Te Si Ni Fe Ti Se
  ISTP  -  ISTj       -  Ti Se Ni Fe Te Si Ne Fi
  ESTP  -  ESTp       -  Se Ti Fe Ni Si Te Fi Ne
  ISFP  -  ISFj       -  Fi Se Ni Te Fe Si Ne Ti
  ESFP  -  ESFp       -  Se Fi Te Ni Si Fe Ti Ne
  ENTJ  -  ENTj       -  Te Ni Se Fi Ti Ne Si Fe
  INTJ  -  INTp       -  Ni Te Fi Se Ne Ti Fe Si
  ENFJ  -  ENFj       -  Fe Ni Se Ti Fi Ne Si Te
  INFJ  -  INFp       -  Ni Fe Ti Se Ne Fi Te Si
  ESTJ  -  ESTj       -  Te Si Ne Fi Ti Se Ni Fe
  ISTJ  -  ISTp       -  Si Te Fi Ne Se Ti Fe Ni
  ESFJ  -  ESFj       -  Fe Si Ne Ti Fi Se Ni Te
  ISFJ  -  ISFp       -  Si Fe Ti Ne Se Fi Te Ni  


I hope this helps!