A correlation design will have two scale variables. A survey is the name for a type of research strategy - it might be a questionable, or interviews or focus groups. Even questionnaires can contain a variety of different question types. This can lead to a wide variety of data analysis designs, including a correlation.
Correlation, ex post facto, experimental (quasi/true), descriptive (though questionable) and survey are considered to be key quantitative designs. Correlation is unique because it tests the relationship between two variables, survey design is considered to be a simple and general design in quantitative paradigm. In most journals, they don't publish studies that are a product of survey designs. If you adopt a survey design, you just want to find out what is happening.
Therefore, quantitative surveys should make use of closed ended questions and not interviews or focus group etc. However, there are some surveys that are undertaken with open ended questions ie interviews and focus group which make them to be qualitative surveys.
The other difference is on analysis, surveys mainly make use of simple descriptive analysis methods . Correlation make use of measures of association (spearman, Pearson Kendall etc)
It is really a stupid distinction ( false antithesis) that is made in some textbooks (e.g.https://www.sagepub.com/sites/default/files/upm-binaries/57732_Chapter_8.pdf)
A survey is one way of collecting information which may be in a quantitative form.
A correlation is one form of analysis for quantitative data; its a very simple technique and generally you get more out of a model-based approach even for 'simple' problems.
On quantitative work and especially modelling you might want to have a look at
Survey make use of descriptive analysis. While, Correlation make use of measure of associate, e.g. (Spearman, Kendall and Pearson. Survey is considered to be a simple and general design, while Correlation test the difference between two or more variables.
As many have already said, a correlation study is done to determine the relationship between 2 or more variables. It's used often when researchers are unable to control the variables. It does not establish cause and effect, even though sometimes you will see that assumption. One of the interesting examples is the positive relationship between ice cream consumption and drownings. Ice cream consumption doesn't cause drownings, they both are related to another variable - warm weather.
Surveys are generally descriptive studies of phenomena. How people have contracted COVID-19 is an example of descriptive information.
Here's the tricky part about surveys and correlational studies. If your sample is large enough in your descriptive survey, you might be able to correlate data on two different items that were collected. If in a COVID-19 survey, age and incidence were collected a researcher could correlate the incidence of COVID-19 with age.
Correlation investigates the relationship between two or more variables without the researcher manipulating any of them while survey is a data collection method which mostly deals with descriptive data.