First thing, Geometric Brownian motion do not have independent increments. It is only Wiener process or Brownian motion that have independent increment. Under GBM, the increments of process (assume stock prices) show markovian property. It means that changes in the process depend on the current price level. In layman terms, the magnitude of change in stock price atleast depend upon existing price level. The probability that stock price will change by \$5 will be different when stock price is \$100, and when stock price is just \$20.
Same argument also hold true with Ornstein Uhlenbeck. The detail answer of why increments under Ornstein Uhlenbeck are not random is given on Math S.E. (Note: The link was first cited by Leon in his answer.)
Now, your question is:
What are the most common stochastic process used in quant finance that
have dependent increments, i.e. $X(t+h)−X(t)$ not independent
To model such stochastic process where increments are not random, you can use ARMA class of models. They are very common in finance and used extensively to model stock return and other process(like interest rate, GDP, etc.) too.