Statistical significance refers to whether the observed effect is larger than we would expect by chance, i.e. can we reject the null hypothesis that there is no effect. This is what is typically addressed by p-values associated with T-tests or ANOVAs etc.
Practical significance is about whether we should care/whether the effect is useful in an applied context. An effect could be statistically significant, but that doesn't in itself mean that it's a good idea to spend money/time/resources into pursuing it in the real world. The truth is that in most situations (at least in psychology), the null hypothesis is never true. Two groups will almost never be *exactly* the same if you were to test thousands or millions of people. That doesn't mean that every difference is interesting.This is usually associated with effect size measures (e.g. Cohen's d; which has criteria for 'small', 'medium' and 'large' effects), but generally will also need to take into account the context of the particular study (e.g. clinical research will have different expectations than personality psychology in terms of what kind of effects can be expected).
For example, say you had an expensive new anti-depressant drug which you want to try out, so you run a trial in which you compare patients on it to patients on other drugs and a placebo. You measure it using a depression questionnaire which gives you a score out of 100, and has cut offs for clinical risk (e.g. <25 is high risk, 25-50 moderate risk, 50-75 low risk, 75+ normal range). You find that your new drug is significantly better than the next best one using a T-test (p=.014), but the average score for the group using the new drug is 31.42 compared to 30.6 in the other drug group. Whilst the drug's effect is *statistically* significant, so we can say it probably isn't due to chance, the increase was very small, and probably doesn't hugely impact the patient's outcome (they're still in the same risk category). Given that the new drug is very expensive, it then doesn't seem very *practical* to pursue it based on the evidence you have.