Differential splicing, or more commonly alternative splicing, is the biological process in which a gene can code for more than one protein, by inclusion or exclusion of different exons on the final messenger RNA. These alternative mRNA are called transcript isoforms, each isoform will result in a different protein.
Differential transcript expression is the statistical analysis where one compares the expression of transcript isoforms between conditions.
The take away is: differential splicing is a biological process, differential transcript expression is an analytical method.
I would say "differential splicing" is for a splicing event (for example, coverage for an exon and/or junction, or frequencies for specific parts of the isoforms). I think JunctionSeq is a nice option for splicing analysis (although, with that definition, DEXSeq, MATS, and MISO are also other popular options).
I would say "differential transcript" is for differences in full-length transcripts. For Illumina data, these would be estimates, where most programs that I can think of (Salmon, cufflinks, etc.) would have some strategy for assigning multi-mapped reads. If you have replicates, I think this can help in situations where there is greater variability in the estimates (for example, some papers show that estimates are less accurate for genes with more isoforms), but "differential gene" analysis using unique reads in unambiguous regions would be another option (using a program like htseq-count).