To add to rightskewed answer:
While it is true that:
Gencode is an additive set of annotation (the manual one done by Havana and an automated one done by Ensembl),
the annotation (GTF) files are quite similar for a few exceptions involving the X chromosome and Y par and additional remarks in the Gencode file (see more at FAQ - Gencode).
What are the actual differences between different annotation databases?
They are a few differences, but the main one for me (and it could be stupid) is
that Refseq is developed by the American NCBI and
the ENSEMBL is mainly developed by the European EMBL-EBI.
Often, labs or people will just start using what is the best known to them (because of a course or workshop) or because they start working with one of the databases with one specific tool and keep with it later.
My lab, for reasons still unknown to me, prefers Ensembl annotations (we're working with transcript/exon expression estimation), while some software ship with RefSeq annotations.
Your lab might be mostly European based people or they might also have read papers like the one from Frankish et al. Comparison of GENCODE and RefSeq gene annotation and the impact of reference geneset on variant effect prediction. BMC Genomics 2015; 16(Suppl 8):S2 - DOI: 10.1186/1471-2164-16-S8-S2
From the Frankish et al. paper paper:
The GENCODE Comprehensive transcripts contain more exons, have greater genomic coverage and capture many more variants than RefSeq in both genome and exome datasets, while the GENCODE Basic set shows a higher degree of concordance with RefSeq and has fewer unique features.
Are there significant differences between them today, or are they, for all intents and purposes, interchangeable (e.g., are exon coordinates between RefSeq and Ensembl annotations interchangeable)?
No. I don't think they are great differences between them as that the global picture should stay the same (although you will see different results if you are interested in a small set of genes). However, they are not directly interchangeable. Particularly as there are many versions of Ensembl and Refseq based on different genome annotations (and those won't be interchangeable between themselves either in most cases).
However, you can easily translate most of your Refseq IDs to ENSEMBL IDs and vice-versa with tools as http://www.ensembl.org/biomart/martview for example (there are devoted libraries/API as well like Biocondutor: biomaRt
 Most as sometimes, they might be annotated in one of the database but haven't (yet) an equivalent in the other.
In fine, even if people tends to keep to what they are used to (and that the annotations are constantly expanded and corrected) depending on the research subject one might be interested in using one database over another:
From Zhao S, Zhang B. A comprehensive evaluation of ensembl, RefSeq, and UCSC annotations in the context of RNA-seq read mapping and gene quantification. BMC Genomics. 2015;16: 97. paper:
When choosing an annotation database, researchers should keep in mind that no database is perfect and some gene annotations might be inaccurate or entirely wrong. [..] Wu et al.  suggested that when conducting research that emphasizes reproducible and robust gene expression estimates, a less complex genome annotation, such as RefGene, might be preferred. When conducting more exploratory research, a more complex genome annotation, such as Ensembl, should be chosen.
 Wu P-Y, Phan JH, Wang MD. Assessing the impact of human genome annotation choice on RNA-seq expression estimates. BMC Bioinformatics. 2013;14(Suppl 11):S8. doi: 10.1186/1471-2105-14-S11-S8.