Symphonies and concertos share a number of characteristics. Both are large scale orchestral forms of music, and both were very popular from the Classical era (1750 onwards). Concertos did form part of earlier Baroque music, but they really came into their own – along with the symphony – a bit later.
The main point of difference is that in a concerto there a featured soloist (or soloists) is given the chance to really stand out. In concert they will stand up front – so not within the instrumental section of the orchestra. In many respects the solo instrument is in something of a battle with the orchestra, largely competing with the ensemble and showing off a bit. In a symphony, while there may be solo passages, the musicians are really all in it together.
Concertos traditionally have three movements, while symphonies have four – though there are plenty that have more, or less. That aside, both follow typical formal musical structures.
The Classical era concerto introduced the ‘cadenza’, which is sort of an improvised ending to the first movement. The soloist is given free rein to really show what they can do – some were composed, but others are left to the performer to realise.