We examine transmit antenna selection with receiver maximal-ratio combining (TAS/MRC) and transmit antenna selection with receiver selection combining (TAS/SC) in multiple-input multiple-output (MIMO) relay networks. Amongst L two-hop relay links, a single relay offering the highest end-to-end signal-to-noise ratio (SNR) is activated. Assuming independent non-identically distributed Weibull fading between the hops, new closed-form asymptotic expressions for the outage probability and the symbol error rate are derived considering NS, NR, and ND antennas at the source, the relays, and the destination, respectively. Based on such expressions, the diversity order and the array gain for M-ary phase shift keying and M-ary quadrature amplitude modulation are analyzed. We highlight that the diversity order of TAS/MRC is the same as TAS/SC. As such, we explicitly characterize the SNR gap between TAS/MRC and TAS/SC as the ratio of their respective array gains. An interesting observation is reached that for equal per-hop SNRs, the SNR gap between the two protocols is independent of L.