Can someone explain the difference between transcript_exon_intron and gene_exon_intron for the getSequence() function in the biomaRt R package?

The documentation (https://bioconductor.org/packages/devel/bioc/manuals/biomaRt/man/biomaRt.pdf) says that transcript_exon_intron gives "gives the full unspliced transcript, that is exons + introns" and gene_exon_intron gives "the exons + introns of a gene".

My guess is that the transcripts are mRNAs that result from alternative splicing, and gene is the wild type or most common transcript?

Ensembl BioMart actually has a nice graphical explanation of this if you use the web interface.  You can see their pictures for the two options below:

gene_exon_intron
transcript_exon_intron

The gene_exon_intron option basically gives you one sequence per gene, containing all the of all exons and introns regardless of splicing.  Even if a gene has two isoforms, and they contain mutually exclusive exons, you won't know about it using this option.  On the other hand transcript_exon_intron returns one sequence per transcript.